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Heterodox Coverings
Chapter 19
Heterodox coverings In this chapter we consider a few generalizations of coverings. In Section 19.1, we consider L-spheres, i.e., we relax the definition of the sphere to allow unions of shells; in Section 19.2, we use spheres of two different radii and in Section 19.3, we restrict all the spheres to be of different radii. In the first two sections we only deal with the perfect case, whereas the third section is devoted to finding optimal coverings. In Section 19.4, we study multicoverings and Section 19.5 deals with a problem related to constant weight coverings. Most of the issues considered here stem from communication problems (see the notes at the end of the chapter), although they are of evident intrinsic interest.
19.1
Perfect coverings by L-spheres
Let L be a fixed subset of [0, n]. For x E IF~, we set S(x)-
{y E I F ~ ' d ( x , y ) E L}.
We call L(x) the L-sphere around x and we say that x covers y if y C L(x). Observe that a vector covers itself if and only if 0 E L. A code C C_ IFn is called an L-covering if
U L(r
IF
cEC
For L-coverings, the sphere-covering bound reads - - cf. (18.2.13) - -
IC]lL(c)l-ICl~ (?)>2". iEL
461
(19.1.1)
462
Chapter 19.
Heterodoz coverings
Another lower bound on ICI is given in Theorem 18.2.14. If equality holds in (19.1.1), C is called a perfect L-covering. Then the sets L(c) for c E C form a partition of IF'~ and [C I - 2 k, ] L ( c ) l - 2'~-k for some k. Such a code is denoted by s k, L). Let us set L - { n - s ~ E L}. We have, for all x E IF'n , L(x) - L(~).
(19.1.2)
Moreover, L(x) M L(~) - 0 if and only if (l E L implies n -
~ ~ L).
(19.1.3)
As for c E C we have 0 E c + C, we always assume 0 E C. Let C - {x E IFn 9 ~ EC}; a code C such that C - C is called self-complementary. If C is not self-complementary we assume that 0 E C and 1 ~ C hold. Define L*
-
{min{s n -
s
s E L}.
Notice that L* C [0, Ln/2J]. L e m m a 19.1.4 I f C - C is a perfect L-covering, then C is a perfect L*covering. Proof.
Use (19.1.2) and (19.1.3).
For a perfect L-covering, we have L e m m a 19.1.5 There is no triangle in IF '~ with side lengths / , s with l, l' in L and c, c' in C.
c'),
P r o o f . Otherwise, we would get a triangle (c, c', x), with c and c' in C, i.e., x E L ( c ) N L(c'), a contradiction. [] The following result is proved in Section 2.4 (Lemma 2.4.6). L e m m a 19.1.6 In only if (i), (ii) and (i) The triangle (ii) a + b + c is (iii) a + b + c <
IFn there exists a triangle with side lengths a, b, c if and (iii) hold. inequalities are satisfied. even. 2n. E]
As an immediate consequence we deduce the following corollaries for a perfect L-covering C.
19.1.
Perfect coverings by L-spheres
463
C o r o l l a r y 19.1.7 If c, c' C C and d(c, c') - 2t for some positive integer t, then t > min{s n - s holds for every s C L. [] C o r o l l a r y 19.1.8 If c C C \ {0} and w(c) is even, then w(c) > 2 maxmin{e, n s
e}.
We are now in a position to prove nonexistence results. Theorem
19.1.9 For a perfect L-covering C that is not self-complementary,
let s C L, 2 ~_ s ~ n/2; then s P r o o f . Suppose the theorem is not true and let s be the smallest integer for which the theorem is violated. Let x be a vector of weight g - 2 and let c be the (nonzero) centre of the (unique) L-sphere containing x, i.e., d(x, c ) = s C L. As 0 C C, by L e m m a 19.1.5 there is no triangle in IF '~, with side lengths t~,s w(c). In view of L e m m a 19.1.6, one of the conditions (i), (ii), (iii) must be violated. By x E L(c) we know the existence of a triangle with side lengths s 2, s w(c). Hence there are only two possibilities: a) either s 2 = s + w(c) and consequently (0, c, x ) i s a fiat triangle; b) or s 2 + s + w(c) = 2n, i.e., s 2 = ( n - s ( n - w(c)) and consequently (0, E, x) is a fiat triangle. (Note that if s = 2, then x = O, s = w(c), so there must be no triangle with side lengths s = 2, s w(c) - s This is possible only if s + s + w(c) -2 n + 2 and w e a r e in case b). I f s 0, i.e., c = x, then w ( c ) = s = s163 and we are in case a). Finally s - 2 and s - 0 is impossible, since s is assumed to violate the theorem.) Let C be the set of codewords covering all the vectors x of weight s 2 in IF'~; C can be partitioned into three classes C1 = {c C C : w(c) _< n/2}, Cr = {c C C : w ( c ) e v e n , w(c) > n/2} and Co = {c C C : w ( c ) o a a , w(c) > n/2}. Then C1 corresponds to case a), C, and Co to case b). We assert that each class contains at most one element. Suppose the contrary; then for the corresponding cl, c2 we have d(cl, c2) = 2t < 2~, in contradiction with Corollary 19.1.7. Indeed, for the first class, w(c~) - s 1 6 3 w(c2) - l - 2 - l ' 2, so by Corollary 19.1.8, w(cl) and w(c2) are odd. This shows that d(cl, c2)is even and d(c~, c2) < 2e. For Ce or Co, we would have w(E1) = s + s - 2 - n, w(E2) - s + s - 2 - n, with the same conclusion for cx and c2. If c C Cx, then (0, c, x) is a flat triangle and c can cover at most n-w(r -
2 -
Chapter 19. Heterodoz coverings
464
vectors of weight e - 2 . Moreover, 0 < w(c) < e - 2 and w(c) is odd by Corollary 19.1.8. If c E Ce or Co, then (0, 3, x) is a flat triangle and c can cover at most
- 2 - ~(e)]
vectors of weight i following inequality:
( ) ( n
<
2
-
-
e-
n
2, with 0 < w(~) < ~ - 2. Hence we are led to the
2-
where 0 < w(el) _< i Ci --
) (o ~(c~) e-
w(el)
+
+ 2
-
~(~~))
2 is odd and n - ( e -
(o e-
2-
~(~o))'
(19.1.10)
2) _< w(ei) < n, for e, - ee and
Co.
The right hand side of (19.1.10)is too small unless e _ 4, w(c~) 1, ~ ( c ~ )
-
=-
1 fo~ s o m e r
r
~ c , in w h i c h r
d(r
r
-
= o~ ~ -
-
2;
furthermore, it is easy to check that, for the same reason, none of the three classes C1, Ce, Co, can be empty. Now this implies that there exists a vector x of weight ~ - 2 which has distance ~ - 3 to el, i.e., g - 3 E L and similarly n - ( ~ - 3 ) E L; this leads to a contradiction: a triangle with side lengths d(cl, c2) - n - 2 or n, ~ - 3 , n - ( t - 3 ) cannot exist by Lemma 19.1.5, however it satisfies conditions (i), (ii) and (iii) of Lemma 19.1.6. [] Note that if C - C, either the unique e covering x is not 1 and the previous proof carries over mutatis mutandis, or e - 1, in which case ~r _ n - ( ~ - 2 ) C L. C o r o l l a r y 1t}.1.11 Let C be a perfect self-complementary L*-covering, ~ C L* 2 < l . Then i - 2 E L* [] )
__
T h e o r e m 19.1.12 For a perfect L-covering C that is not self-complementary, let l E L, n/2 <_ l < n - 2. Then ~ + 2 E L. P r o o f . If C is a perfect L-covering, then C is a perfect L-covering. If C is not self-complementary, then C is not self-complementary. Hence Theorem 19.1.9 implies Theorem 19.1.12. [] By the previous two theorems, the set L for perfect L-coverings which are not self-complementary is determined by four integers - 2 <_ al, a2, bl, b2 <_ n/2, with al, bl odd, a2, b2 even, in the following way" L-{2sl+1"O<2sl+l__al}U{2s2"O_2s2
19.1.
Perfect coverings by L-spheres
465
U { n - 2s3 -- 1" 0 < 2s3 A- 1 < bi} U { n - 2s4" 0 < 2s4 < b2}, which we write L - L1U L2 UL3 UL4. By the previous two corollaries, the same is true for self-complementary codes and L*. From now on, we assume rather naturally that L C [0, n/2], i.e., L3 - L4 - 0. By the previous corollary, for perfect self-complementary L*-coverings, L* is of the form L1 t2 L2. T h e o r e m 19.1.13 If C is a perfect L-covering, with L C_ [O,n/2], ai >_ 1 and a2 >_ O, then lai - a 2 [ - 1, that is, L is the interval [0, max{ai, a2}]. [] In other words, when both Li and L2 are nonempty, L(x) is the "usual sphere". We omit the proof (see [148]). Consequently, the following theorem, which deals with the case when one of the two sets L1 or L2 is empty, enables us to classify the perfect L-coverings, when L C_ [0, n/2]. T h e o r e m 19.1.14 Let e - R - 2[R/2J. Then there exists an s {0, 1, 2, . . . , R}) if and only if there exists an s + 1, k Jr 1, L' - {R, R 2, R - 4 , . . . , e } ) . P r o o f . Let C be an s k , L - {0, 1, 2 , . . . , R}), i.e., a perfect code with covering radius R. For any distinct ei and c2 in C, d(ei, e2) > 2R + 1. Now let C ' - C @ IF (cf. Section 3.2). Obviously, for any distinct e~ and e~ in C', d(c~, c~) equals 1 or is at least 2R + 1. Thus L'(e~) and L'(e~) have an empty intersection. Furthermore the sphere-covering equality for C can be rewritten as
2k+iE iEL'
(n+1)-2'~+1 i
hence C' is a perfect L'-covering. Conversely, let C' be an s + 1, k + 1, L') with e - 0 (respectively, 1). Then the 2"~ even vectors in IFT M are covered by the even (respectively, odd) codewords in C'. Deleting the last component in IF'~+i we get an s k, L). []
C o r o l l a r y 19.1.15 The only perfect L-coverings with L C_ [0, n/2] are the classical perfect codes and those with the following parameters: (2R + 2, 2, {R, R - 2 , . . . , R - 2 [R/23 }),
s
2"~ - m, {1}), ~(24, 13, {1, 3}),
derived from binary repetition, Hamming and Golay codes, respectively.
[]
Chapter 19. Heterodox coverings
466
So, either when L is included in [0, n/2] (or, by (19.1.2), in [n/2, n]) or when the code is self-complementary, we know all the p a r a m e t e r s of perfect L-coverings (note t h a t for a given self-complementary code, various sets L are possible). We now study a case when L is not included in [0, n/2], to give a flavour of m e t h o d s t h a t can be used (first note t h a t the construction of T h e o r e m 19.1.14 yields a n / : ( n + 1, 1, L ' - {n, n - 2 , . . . , n - 2[n/2~}) - {0 '~+1, 0'~1} obtained from the trivial perfect code {0'~}). The
case
L -
[0,1] U { n }
Let us first give an easy existence result. Theorem
2}).
Proof.
1 9 . 1 . 1 6 For m > 2, there exists an E ( 2 m - 2 , 2 m - 2 - m , {0, 1, 2 m Shorten a H a m m i n g code of length 2 TM - 1.
[]
On the other hand, as we now prove, for l - 2 there is no such perfect code: Theorem Proof.
19.1.17
There is no nontrivial s
k, {0, 1, 2, n}).
The sphere-covering equality would give, setting n n 2 + n-
k - r"
2(2 ~ - 2) - 0.
The discriminant 2 ~+3 - 15 can be a nonzero square only if 2~ + 3 - z which implies z 2 - 1
2
(mod3),
(mod 3), r + 3 - 27 and 15 - 2 r+3 - z 2 - (2 ~ - z)(2 "r q- z).
Solving for 7 and z gives only the trivial E(3, 0, [0, 3]) code.
[:]
We conclude with a conjecture. C o n j e c t u r e 1 9 . 1 . 1 8 Let L - L1 U L2 U L3 U L4 be the decomposition of L as before. If L i r 0, for i - 1, 2, 3, 4, then L1 U L2 and L3 U L4 are intervals containing 0 and n, respectively. Before giving some results in the nonbinary case, we remind the reader t h a t Corollary 18.2.15 states t h a t no perfect L-covering exists for L - [ ( n w ) / 2 , (n + w)/2].
19.2.
Perfect coverings by spheres of two radii
The
nonbinary
467
case
Here, the situation is simpler and perfect L-coverings are classified as follows. T h e o r e m 19.1.19 For q > 2, the only possible perfect L-coverings occur for L - [0, R], i.e., are the classical ones. The proof follows from two easy lemmas (cf. Lemmas 19.1.5 and 19.1.6). L e m m a 19.1.20 If C is a perfect L-covering, there is no triangle in 7/qn with side lengths t, ~t, d(c, c'), with ~, ~' in L and c, c r in C. [] L e m m a 19.1.21 In 7/q,9t q > 2, there ezists a triangle with side lengths a, b, c if and only if the triangle inequalities are satisfied. [] P r o o f of T h e o r e m 19.1.19 First note that there is no l in L with l > n/2, because a triangle with side lengths ~, ~, w(c) (with c E C), possible by Lemma 19.1.21, would violate Lemma 19.1.20; hence L C [0, n/2]. Suppose that L is not an interval [0, R]. Let x be such that w(x) - min{i" i ~ L} andmmax{i" i E L). T h e n x ~ L(0) and there is a c i n C f o r which x E L(c). Now w(c) < m + w(x) < 2m, so there is a triangle with side lengths m, m, w(c) (by Lemma 19.1.21), which is impossible, by Lemma 19.1.20. []
19.2
Perfect coverings by spheres of two radii
Let us consider another generalization of perfect codes, where the spheres have different radii. Let C - Ul<~ 4, there ezists a P(2 TM, {2, 1}, {22"-2m, 22"~-m-1 - 22~-2"~}). P r o o f . For m even, m > 4, there exists a punctured Preparata code 7~,~ with parameters (2"~-1, 22"~-2"~, 5)3, which is a subcode of the [2m-1, 2 " ~ - m - 1 , 3] Hamming code 7~,,~ (cf. Theorem 2.6.5). Furthermore, d(x, 7 ~ ) - 3 if and only if x E 7~,,~ \ ~P,~,
468
Chapter 19. Heterodox coverings
since
[7)*[" V( 2m
-
1, 2 ) + [ ~ , ~ [ - [ ' P * [ - 22'~-1
Setting C2 - 7~*~{0}, R2 - 2, C1 - (7/,~\7~)(9{1}, R1 - 1, it is again easily checked by counting that we obtain a partition of IF 2"~ with the announced parameters. []
T h e o r e m 19.2.2 There exists a P(15, {3, 1}, {32,896}). P r o o f . Take for C1 the [15, 5, 7]5 punctured Reed-Muller code 7~A4"(1, 4) with R1 - 3. There are 28.25 vectors at distance 5 from 7~A4" (1, 4), the socaned "bent" functions (see Notes on Section 9.2), and they form a (15,896, 3) code C2. Set R2 - 1 and conclude by noticing that 3
1
i=0
i=O
-
21s.
T h e o r e m 19.2.3 There exists a P ( l l , {3, 1}, {2, 132}). P r o o f . There exists a Steiner system S(4, 5, 11) (see Example 3.3.3 where it is used to prove that K ( l l , 1) < 192). Together with its complement S, this gives an (11,132, 3) code C1 with the property that for all x 6 IF 11 such that 4 _ w(x) _< 7,-there exists a unique codeword e 6 C1 with d(x, e) _< 1. Take C2 - {011, 111}, R2 - 3 to obtain the existence of P ( l l , {3, 1}, {2,132}). []
T h e o r e m 19.2.4 For s > 4, a P(2s + 3, {s - 1, 1}, {K1, K2}) exists only if g l - 2, K 2 - 2(28~3)/(s + 1). P r o o f . Suppose there exists a P ( 2 s + 3 , { s - l , 1}, {K1, K2}). One can always centre a sphere of radius s - 1 at 0. Suppose K1 >_ 3. Then there would be at least two other spheres Bs_x(x) and Bs-I(y). Now B s _ l ( 0 ) A B s - l ( x ) 0, B s - l ( 0 ) n B , _ l ( y ) - 0, implies w(x) >_ 2 s - 1, w(y) >_ 2 s - 1. For s > 4, B s - l ( x ) and B s - l ( y ) would intersect in 1, the all-one vector of length 2s + 3, which is impossible. Hence K1 _< 2. Suppose now K1 - 1, so the only sphere of radius s - 1 is centred at 0. Call Jr/ the number of spheres whose centres have weight i. Elements of weight s in IF 2~+3 are contained in spheres with centres of weight s + 1. Every such sphere contains s + 1 elements of weight s, so .h,+l - (2s+3)/(s + 1). The same method then shows that A,+2
-
A,+I,
A,+3 - A,+4
-
0.
19.2. Perfect coverings by spheres Of two radii
469
Now spheres with centres of weight s + 5 must contain all elements of weight s + 4 and at most the total number of elements of weight s + 6, hence
~t~+~
_
§
\ ~ - 1 ] / ( ~ + 5)
and
A~+~ <
/ ( ~ - 2) -
+-----~
/(~ +
5),
a contradiction. Finally K1 - 2. Now the sphere-covering equality
K1V(2s + 3, s - 1 ) + K2V(2s + 3, 1 ) - 22'+3 gives K 2 - 2(2"+3] /(s -t- 1). k $ ! Puncturing sometimes allows to obtain new perfect codes from existing ones. Let us mention the following constructions: T h e o r e m 19.2.5 Let P(n, {R1, R2}, {K1, K2}) be such that
K1
()
n-1 R1
-K2
(. 1) R2
;
if moreover any two adjacent spheres of the same radius have the same i-th component, then puncturing on the i-th position yields two perfect 2-radius codes: P(n - 1, {R1 - 1, R2}, { g l , g2}), P(n - 1, {R1, R2 - 1}, {K1, K2}) (two spheres of radius Rj are adjacent if the distance between their centres is
2Rj + 1). Proof.
Straightforward.
E x a m p l e 19.2.6 Viewing the Golay code as a P(23, {3, 3}, {211,211})gives rise to a P(22, {3, 2}, {211, 211}). In fact a further puncturing gives perfect a 3-radius code P(21, {3, 2, 1}, {21~ 211, 21~
[]
Chapter 19.
470
19.3
Heterodox coverings
Coverings by spheres all of different radii
In this section we consider the problem of coverings by spheres when no two equal radii are allowed. Namely, we allow the occurrence in a covering of at most one sphere of each of the radii 1, 2 , . . . , n. The problem is to find a covering minimizing the m a x i m u m radius of a sphere used in the covering. Evidently, one sphere of radius n is enough to cover all the space. As well, two spheres of radii In~21 - 1 and In~21 centred at two complementary vectors suffice. Thus, an evident upperbound on the m a x i m u m radius of a sphere involved in the covering is In~21 . Surprisingly enough we cannot do better. Let us rephrase the problem more precisely. Let I - {R~,..., R~}, R~ < R2 < . . . < Rm, be a subset of { 1 , . . . , n } , m < n and C - { e l , . . . , Cm}, be a collection of centres. Assume that m
[.J
- IF"
j=l
Given n we want to construct a covering C with minimal P ~ . The m i n i m u m of Rm over all possible C is denoted by p(n). L e m m a 19.3.1 Let & - ( a i ) - ( a i , j ) , i - 1 , . . . , n , j 1 , . . . , n , be a matrix with entries 4-1. Then there exists a 4-1-vector y+ - ( y ~ ) such that
I E Y~a',jl<2i l <_j<_,~
for i - - 1 , . . . , n. Proof. For each i, consider the homogenous system of n equations zi = n ~ j = 1 ai,jxj in the n real variables x l , . . . , xn. Set x - ( e l , . . . , xn) and y - ( y l , . . . , yn), where the yi's belong to { - 1 , 0, +1}. A trivial solution to A x T - On is y - On. At step one, relax, say, the last equation: delete the last row of .4, and denote by _4,1 the resulting matrix; then the set of solutions to A1x T - 0n-1 is a vector space of dimension at least one, thus containing a line through the origin. This line hits the real cube [ - I , 1]n on a face; after possible reordering, we can assume that it is the n-th face, of equation xn equal to, s a y , - 1 . Set y - (On - I ~ - 1 ! and repeat this procedure n - I times. T h a t is, before step t + ~ y - ( , , ] y n - t + 1 , . . . , yn) where y n - t + l , . . . , yn are t +1 elements determined by the previous moves; at step t + 1, relax the ( n - t ) - t h equation, determine the space of solutions to the system of n - t - 1 equations in n - t unknowns, find a face where it intersects the cube [ - I , 1]n-t. This fixes a new component of y, say S/n-t, and defines y - ( O n - t - 1 1 y n _ t , . . . , y n ) . Denote the final 4-1 result, obtained after step n, by y+ and consider the sum ~ 1 < j < ~ y~:aij. From the description of the
19.3.
Coverings by spheres all of different radii
471
process, this sum remains 0 until we reach step n - i + 1. During step n - i + 1, the absolute value of this sum increases by at most one, and by at most two during the i - 1 remaining steps. So [ ~ i < j < , ~ Y~ai,Jl < 1 + 2 ( i - 1) < 2i. []
Now we are in position to prove Theorem
19.3.2
,o(,-,)- r,-,/2l. P r o o f . Assume the contrary, i.e., that we have a covering C - { c i , . . . , c,,~}, with R ~ < ~n/2] - 1. W i t h o u t loss of generality, we can assume that R1 - 1, R2- 2,..., Rm-m, with m ~ n / 2 1 - 1 . Let c • be the vector obtained from e by changing O's to l's and l's to - l ' s , and let E be the complement of c. Construct a m a t r i x / k having as first rows the words era_i,• . . . , c~, the • remaining rows being arbitrary. By L e m m a 19.3.1, there is a + l - v e c t o r y such that l_
i - 1 , . . . , m - 1. Since ~l<_j n/2 - m + i and d(ci, y) > n/2 - m + i. Since n / 2 - m + i > i - Rq, neither y nor y belong to BR,(ei), for i - 1 , . . . , m - - 1. Evidently, one of the two distances d(em, y) or d(e,n, y) is strictly greater than In/21 - 1. Let it be d(e, y): this means that y is not covered by C. On the other hand, Br~/2](0 ) and Bb~/2]_l(1 ) provide such a covering. []
This problem has an application to broadcasting in the n-dimensional cube. Let 2'~ processors be connected as in the n-dimensional cube, i.e., each processor is assigned a different binary n-tuple and two processors are connected if the corresponding binary vectors are at Hamming distance one. Now, to broadcast a message in the n-cube, it is allowed at each round to provide it to exactly one of the processors; in parallel, a processor that already received the message sends it to its neighbours. Thus, after the first round, one processor, say pi, knows the message; after the second round, pi, its neighbours and some other processor p2 know the message, and so on. The question is: how m a n y rounds are needed to deliver the message to all processors? Theorem 19.3.2 gives the answer, namely In/2] + 1, and the strategy is very easy: just send the message to two antipodal nodes during the first two rounds and after this, ... relax.
Chapter 19. Heterodoz coverings
472
19.4
M u l t i c o v e r i n g radius
In this section we study codes with the property that for every s-tuple of vectors in the Hamming space there is a codeword which is close to all of them. This leads to the concept of multicovering radius. D e f i n i t i o n 19.4.1 Let s >_ 1 be fized.
The multicovering radius of a code C C IFn is the smallest integer R, - Rs(C) such that for every s-tuple (xl, x 2 , . . . , xs), where X1, X 2 , . . . , Xs E IFn, there is a codeword c E C such that d(c, xi) < Rs for all i - 1 , 2 , . . . , s . When s Theorem
1, the definition of course reduces to that of covering radius.
19.4.2 If C is an (n, K) code, then Rs(C) > n - t(n, s).
P r o o f . Recall that t(n, s ) i s the smallest covering radius among all (n, s) codes. Take as the vectors x t, x 2 , . . . , xs the codewords of a binary code of length n with covering radius t(n, s). If y C IF'*, then y is within distance t(n, s) from at least one of these s vectors, say xi, and therefore d(y, xi) >_ n - t(n, s). In particular, any codeword in C is at distance at least n - t(n, s) from at least one xj. []
C o r o l l a r y 19.4.3 If s > 2, then the multicovering radius Rs of any code of
The following theorem shows that determining Rs(IF '~) for all s and n is equivalent to determining the values of K(n, R) for all n and R. Theorem
K(n,n-t-
10.4.4 The quantity Rs (IF'~) equals the smallest integer t such that 1) > s.
P r o o f . We see that R,(IF '~) _< t if and only if for every s-tuple of vectors, say x t , x 2 , . . . , x , there is a point c E IFn such that d(xi, c) ___ t for all i; or equivalently that for every s vectors x t , . . . , x, there is a vector c C IF'~ such that d(xi, 3) >_ n - t. But this is equivalent to saying that K ( n , n - t - 1) > s. []
Determining Rs(IF") is an essential problem also from the point of view of studying the multicovering radius of other codes as can be seen from the following immediate result.
19.5. Perfect coverings of a sphere and constant weight coverings
473
Theorem
19.4.5 For every code C, R,(IF '~) ~ R,(C) ~_ R , ( I F n ) + R ( C ) .
Theorem
19.4.6 R2 (7/m) - 2 m- t.
[]
P r o o f . Let n -- 2' ~ - 1. By Corollary 19.4.3, R2(7/m) >_ r~/21 = 2 ~ - , . it remains to show that R2(7/m) _< (n + 1)/2. Let x, y E IF'~ be arbitrary. If d(x, y) <_ n - 1, then let z C IF '~ be any vector such that d(z, x) _< ( n - 1 ) / 2 and d(z, y) <_ ( n - 1 ) / 2 . Because 7/,,~ has covering radius one, there is a codeword c C 9/,,~ such that d(e, z) __ 1. Then d(c, x) _ (n + 1)/2 and d(c, y) _ (n + 1)/2. Second, assume that d(x, y) = n, i.e., y = ~, and show that there is a codeword c C 7/,~ such that d ( x , c ) E { ( n - 1 ) / 2 , ( n + 1)/2}. Then also d(y, c) = n - d ( x , c) e { ( n - 1)/2, (n + 1)/2}. Let I-I be a parity check matrix of 7/,~ and denote s = t t x T. It is sufficient to show that there is a vector u E IF ~ of weight (n - 1)/2 or (n + 1)/2 such that H u T = s, because then x+uEC. Take I_i_ ( 0 vl
0 v2
... ...
0 v(~-t)/2
1 0
1 vl
1 v2
... ...
1 v(,~-t)/2
)
where the vectors vi run through all the nonzero binary ( m - 1)-tuples. Clearly, the columns on the left hand side add up to 0, and the same is true for the right hand side. If the column s occurs on the right hand side, then all the columns on the right hand side except s add up to s, and we are done. If s appears on the left hand side, then all the columns on the left hand side except s ~aa ~p to ~, b~t now t h r ~ just ( ~ - 3)/2 r ~o w~ have to replace one of them, say (0, vi) T, by the two columns (1, vi) T and (1, 0) T from the right hand side, and we are done. []
19.5
Perfect coverings of a sphere stant weight coverings
and
con-
Consider now the sphere B~(0) of radius r centred at 0 in IF'~. We search for the minimum number of spheres necessary to partition B~(0) into smaller spheres. In other words, we are looking for the minimum number of elements ci C Br(0) which cover B~(0), where each ci covers the sphere Br,(c,) (r, ~ ~ - ~(~)). Consider S~(0), the surface of S , ( 0 ) . Then a partition of Br(0) contains a covering C of St(0), whose elements are the centres of the spheres tangent
Chapter 19. Heterodox coverings
474
to S~(0). Let C~ be the set of elements of C of weight w, r >__ w > 0. Then C~ is a constant weight code and, unless it has only one element, its minimum distance d is even, with d/2 _> r - w + 1 (note also that 2w _ d, i.e., w ___ (r + 1)/2). Using the notation of Section 2.7, we see that [C~ I >__ K(n, w, r, r - w). On the other hand, by Lemma 12.6.8,
IC~o,< n ( n - 1 -
w
w-
(n-w+d/2) d/2
1"'"
) ....
(19.5.1)
Since C~ covers A~o points of S~(O), where
inequality (19.5.1) implies that A~ r ( r - 1 ) . . . ( r - ( r - w) + 1) (~) -< ( n - 2 w + r ) : . . ( n - 2 w + r - ( r - w ) + l ) " Asymptotically, for r fixed" A~ - O(n~). That is, A~ - O(n ~-1) for w-r-landA~-O(n ~-2) for O < w < r - l. So ~ =r-1 l A, - O(n ~- 1) and there are 12(n~) isolated points (spheres of radius zero)in S~(0)N C. Non-asymptotically, supposing that r _~ n/3: A
(:) < ~)
m
~'~?" -- TM
where # - r / ( n - r ) . The sum of the above for 0 < w < r is at most # / ( 1 - # ) . So the proportion of isolated points in a covering of S~(0) is at least 3r
1 - 2#
n-
1-#
n-2r
In other words, C contains at least
(:)~
n - 2r
isolated points. Loosely speaking: If you want to partition a sphere into smaller ones, you have to take a lot of spheres of radius 0 (isolated points). The nontrivial ones cover only a negligible amount of the total volume. Notice that this problem is related to that of perfect p-radius codes dealt with in Section 19.2: namely, by adding the sphere B n - r - l ( 1 ) to a perfect covering of Br(O), we get a perfect covering of IF'~ by spheres with different radii. The following classical theorem in graph theory now solves the problem of covering $3 with elements of $2.
19.6.
Notes
475
T h e o r e m 19.5.2 If the graph G has n vertices and m edges, where m > n2/4, then G contains a triangle. P r o o f . Let G : (V, E) be a graph with no triangle, and let d(u) denote the degree (i.e., the number of adjacent vertices) of a vertex u E V. Then two adjacent vertices have disjoint sets of neighbours, and d ( u ) + d(v) <_ n for every edge {u, v} e E. Trivially ~ v d(v) = 2m and therefore summing over all edges, we get
4m 2
ran>_ Z v6.v
n
vEv
proving our claim.
[:]
On the other hand, a bipartite graph G - (V,E) with V - V1 u V2, E - Vl • V2 shows that there exist graphs with n vertices, [n/2J [n/2] edges and no triangle. Now let C C_ $2 be a 1-covering of $3 in IF'~. We consider the graph G(C) - (V,E) with V - { 1 , 2 , . . . , n } and E - {{i,j} 9e, + e j E C}. This graph has the following property: G(C) contains at least one edge of any triangle T in the complete graph on V. Indeed, otherwise the element of $3 corresponding to T would not be covered by C. In other words, the complementary graph of G(C) is triangle-free, i.e., contains at most n2/4 edges by Theorem 19.5.2. Hence C has size at least ( ~ ) - n2/4 and this lower bound can be achieved"
Ivy1- [Ivl/2J, Iv2l- FlVl/2]
C o r o l l a r y 19.5.3 For all n, K(n, 2, 3, 1 ) - [n(4-2) ~.
More generally, T h e o r e m 19.5.4 If n - t ( r - 1 ) + r0 with 0 < ro < r g ( n , 2, r, r - 2) - (r - 1) (t2) + tro.
19.6
1, r > 2, then [:]
Notes
w L-packings were introduced by Karpovsky [363], where their practical significance - correction or detection of physical failures in networks - is described. In [363] some perfect L-codes are given, as well as an analogue of Lloyd's theorem (cf. Theorem 11.2.1). See also Deza, Karpovsky and Milman
[199].
476
Chapter 19. Heterodoz coverings
Corollaries 19.1.7, 19.1.8 and 19.1.15, Theorems 19.1.9, 19.1.12, 19.1.13 and 19.1.14 are by Cohen and Frankl [148]. Theorem 19.1.16 is due to Karpovsky [363]. Theorem 19.1.17 is by Laurent (unpublished). Reliquet (unpublished) has used methods similar to those of Cohen and Frankl [148] to rule out the existence of nontrivial s k, L) for L - [0, 2t + 1] U {n} and L - {1, 3 , 5 , . . . , 2 t + 1, n} when 3 < 2t + 1 _ n/2. Using earlier results proved by Laurent (unpublished), we get that nontrivial perfect L-coverings with L - [O,l] U {n} can only exist if ~ - 1 or ~>6. The nonbinary case is treated by Cohen and Frankl [149]. w Applications of perfect p-radius codes to source coding and unequal protection of messages are described by Montaron and Cohen [495]. Theorems 19.2.1, 19.2.2, 19.2.3 and 19.2.5 as well as Example 19.2.6 are by Cohen and Montaron [167] [495]. Theorem 19.2.4 is due to Cohen and Frankl [150]. More can be said about Theorem 19.2.1; first, another construction, with the same parameters as, but nonisomorphic to, that of Theorem 19.2.1, was obtained by van den Akker, Koolen and Vaessens [13], by letting C2 7~m, R2 - 2, C1 - ?/m \ 7~,~, R1 - 1, where ^ denotes extension. Now let us call two codewords c E C{, c' E Cj adjacent if d(c, c') R~ + Rj + 1. Then to a code C one can associate in an obvious way a graph F(C) whose set of vertices is C, C being caned bipartite if r ( c ) i s , with C1 and C2 being independent sets (i.e., there are no edges within C1 and C2). For example, the code C - (?/,~ \7~,,~)u7~,,~ is bipartite, since d(c, c') >_ 6, (respectively, 4) if c, c' E C2 (respectively, C~). But the code of Theorem 19.2.1 is not bipartite. Based on the fact that a nearly perfect code which is not perfect and has covering radius at least 3, has the same parameters as a punctured Preparata code (see Theorem 11.6.3), it is proved in [13] that a bipartite 2-radius (R, 1)perfect code with R > 2 necessarily has parameters given by Theorem 19.2.1. On this topic, see also Zinoviev and Katsman [711]. w Lemma 19.3.1 is due to Beck and Spencer [63]. Its use for finding the best strategy for transmitting in the n-cube is by Alon [15]. w The problem discussed in this section is due to Klapper [377] and arose from investigations concerning the cryptanalysis of stream ciphers [376]. Corollary 19.4.3 and Theorem 19.4.6 are from [377], where a more detailed discussion about the multicovering radius and, e.g., its behaviour in various constructions, can be found. w Theorem 19.5.2 is due to Mantel [468]. A result by Turs [656] settles the more general case of covering S~ with a subset of $2. On the
19.6.
No~es
477
subject of perfectly covering a sphere, see Fachini and K;Srner [227] for a proof that no sphere can be partitioned into two spheres, i.e., there is no perfect code in IF'~ with three spheres of any, not necessarily equal, radii. See Etzion, Wei and Zhang [226] for a survey on constant weight coverings.
Heterodox coverings In this chapter we consider a few generalizations of coverings. In Section 19.1, we consider L-spheres, i.e., we relax the definition of the sphere to allow unions of shells; in Section 19.2, we use spheres of two different radii and in Section 19.3, we restrict all the spheres to be of different radii. In the first two sections we only deal with the perfect case, whereas the third section is devoted to finding optimal coverings. In Section 19.4, we study multicoverings and Section 19.5 deals with a problem related to constant weight coverings. Most of the issues considered here stem from communication problems (see the notes at the end of the chapter), although they are of evident intrinsic interest.
19.1
Perfect coverings by L-spheres
Let L be a fixed subset of [0, n]. For x E IF~, we set S(x)-
{y E I F ~ ' d ( x , y ) E L}.
We call L(x) the L-sphere around x and we say that x covers y if y C L(x). Observe that a vector covers itself if and only if 0 E L. A code C C_ IFn is called an L-covering if
U L(r
IF
cEC
For L-coverings, the sphere-covering bound reads - - cf. (18.2.13) - -
IC]lL(c)l-ICl~ (?)>2". iEL
461
(19.1.1)
462
Chapter 19.
Heterodoz coverings
Another lower bound on ICI is given in Theorem 18.2.14. If equality holds in (19.1.1), C is called a perfect L-covering. Then the sets L(c) for c E C form a partition of IF'~ and [C I - 2 k, ] L ( c ) l - 2'~-k for some k. Such a code is denoted by s k, L). Let us set L - { n - s ~ E L}. We have, for all x E IF'n , L(x) - L(~).
(19.1.2)
Moreover, L(x) M L(~) - 0 if and only if (l E L implies n -
~ ~ L).
(19.1.3)
As for c E C we have 0 E c + C, we always assume 0 E C. Let C - {x E IFn 9 ~ EC}; a code C such that C - C is called self-complementary. If C is not self-complementary we assume that 0 E C and 1 ~ C hold. Define L*
-
{min{s n -
s
s E L}.
Notice that L* C [0, Ln/2J]. L e m m a 19.1.4 I f C - C is a perfect L-covering, then C is a perfect L*covering. Proof.
Use (19.1.2) and (19.1.3).
For a perfect L-covering, we have L e m m a 19.1.5 There is no triangle in IF '~ with side lengths / , s with l, l' in L and c, c' in C.
c'),
P r o o f . Otherwise, we would get a triangle (c, c', x), with c and c' in C, i.e., x E L ( c ) N L(c'), a contradiction. [] The following result is proved in Section 2.4 (Lemma 2.4.6). L e m m a 19.1.6 In only if (i), (ii) and (i) The triangle (ii) a + b + c is (iii) a + b + c <
IFn there exists a triangle with side lengths a, b, c if and (iii) hold. inequalities are satisfied. even. 2n. E]
As an immediate consequence we deduce the following corollaries for a perfect L-covering C.
19.1.
Perfect coverings by L-spheres
463
C o r o l l a r y 19.1.7 If c, c' C C and d(c, c') - 2t for some positive integer t, then t > min{s n - s holds for every s C L. [] C o r o l l a r y 19.1.8 If c C C \ {0} and w(c) is even, then w(c) > 2 maxmin{e, n s
e}.
We are now in a position to prove nonexistence results. Theorem
19.1.9 For a perfect L-covering C that is not self-complementary,
let s C L, 2 ~_ s ~ n/2; then s P r o o f . Suppose the theorem is not true and let s be the smallest integer for which the theorem is violated. Let x be a vector of weight g - 2 and let c be the (nonzero) centre of the (unique) L-sphere containing x, i.e., d(x, c ) = s C L. As 0 C C, by L e m m a 19.1.5 there is no triangle in IF '~, with side lengths t~,s w(c). In view of L e m m a 19.1.6, one of the conditions (i), (ii), (iii) must be violated. By x E L(c) we know the existence of a triangle with side lengths s 2, s w(c). Hence there are only two possibilities: a) either s 2 = s + w(c) and consequently (0, c, x ) i s a fiat triangle; b) or s 2 + s + w(c) = 2n, i.e., s 2 = ( n - s ( n - w(c)) and consequently (0, E, x) is a fiat triangle. (Note that if s = 2, then x = O, s = w(c), so there must be no triangle with side lengths s = 2, s w(c) - s This is possible only if s + s + w(c) -2 n + 2 and w e a r e in case b). I f s 0, i.e., c = x, then w ( c ) = s = s163 and we are in case a). Finally s - 2 and s - 0 is impossible, since s is assumed to violate the theorem.) Let C be the set of codewords covering all the vectors x of weight s 2 in IF'~; C can be partitioned into three classes C1 = {c C C : w(c) _< n/2}, Cr = {c C C : w ( c ) e v e n , w(c) > n/2} and Co = {c C C : w ( c ) o a a , w(c) > n/2}. Then C1 corresponds to case a), C, and Co to case b). We assert that each class contains at most one element. Suppose the contrary; then for the corresponding cl, c2 we have d(cl, c2) = 2t < 2~, in contradiction with Corollary 19.1.7. Indeed, for the first class, w(c~) - s 1 6 3 w(c2) - l - 2 - l ' 2, so by Corollary 19.1.8, w(cl) and w(c2) are odd. This shows that d(cl, c2)is even and d(c~, c2) < 2e. For Ce or Co, we would have w(E1) = s + s - 2 - n, w(E2) - s + s - 2 - n, with the same conclusion for cx and c2. If c C Cx, then (0, c, x) is a flat triangle and c can cover at most n-w(r -
2 -
Chapter 19. Heterodoz coverings
464
vectors of weight e - 2 . Moreover, 0 < w(c) < e - 2 and w(c) is odd by Corollary 19.1.8. If c E Ce or Co, then (0, 3, x) is a flat triangle and c can cover at most
- 2 - ~(e)]
vectors of weight i following inequality:
( ) ( n
<
2
-
-
e-
n
2, with 0 < w(~) < ~ - 2. Hence we are led to the
2-
where 0 < w(el) _< i Ci --
) (o ~(c~) e-
w(el)
+
+ 2
-
~(~~))
2 is odd and n - ( e -
(o e-
2-
~(~o))'
(19.1.10)
2) _< w(ei) < n, for e, - ee and
Co.
The right hand side of (19.1.10)is too small unless e _ 4, w(c~) 1, ~ ( c ~ )
-
=-
1 fo~ s o m e r
r
~ c , in w h i c h r
d(r
r
-
= o~ ~ -
-
2;
furthermore, it is easy to check that, for the same reason, none of the three classes C1, Ce, Co, can be empty. Now this implies that there exists a vector x of weight ~ - 2 which has distance ~ - 3 to el, i.e., g - 3 E L and similarly n - ( ~ - 3 ) E L; this leads to a contradiction: a triangle with side lengths d(cl, c2) - n - 2 or n, ~ - 3 , n - ( t - 3 ) cannot exist by Lemma 19.1.5, however it satisfies conditions (i), (ii) and (iii) of Lemma 19.1.6. [] Note that if C - C, either the unique e covering x is not 1 and the previous proof carries over mutatis mutandis, or e - 1, in which case ~r _ n - ( ~ - 2 ) C L. C o r o l l a r y 1t}.1.11 Let C be a perfect self-complementary L*-covering, ~ C L* 2 < l . Then i - 2 E L* [] )
__
T h e o r e m 19.1.12 For a perfect L-covering C that is not self-complementary, let l E L, n/2 <_ l < n - 2. Then ~ + 2 E L. P r o o f . If C is a perfect L-covering, then C is a perfect L-covering. If C is not self-complementary, then C is not self-complementary. Hence Theorem 19.1.9 implies Theorem 19.1.12. [] By the previous two theorems, the set L for perfect L-coverings which are not self-complementary is determined by four integers - 2 <_ al, a2, bl, b2 <_ n/2, with al, bl odd, a2, b2 even, in the following way" L-{2sl+1"O<2sl+l__al}U{2s2"O_2s2
19.1.
Perfect coverings by L-spheres
465
U { n - 2s3 -- 1" 0 < 2s3 A- 1 < bi} U { n - 2s4" 0 < 2s4 < b2}, which we write L - L1U L2 UL3 UL4. By the previous two corollaries, the same is true for self-complementary codes and L*. From now on, we assume rather naturally that L C [0, n/2], i.e., L3 - L4 - 0. By the previous corollary, for perfect self-complementary L*-coverings, L* is of the form L1 t2 L2. T h e o r e m 19.1.13 If C is a perfect L-covering, with L C_ [O,n/2], ai >_ 1 and a2 >_ O, then lai - a 2 [ - 1, that is, L is the interval [0, max{ai, a2}]. [] In other words, when both Li and L2 are nonempty, L(x) is the "usual sphere". We omit the proof (see [148]). Consequently, the following theorem, which deals with the case when one of the two sets L1 or L2 is empty, enables us to classify the perfect L-coverings, when L C_ [0, n/2]. T h e o r e m 19.1.14 Let e - R - 2[R/2J. Then there exists an s {0, 1, 2, . . . , R}) if and only if there exists an s + 1, k Jr 1, L' - {R, R 2, R - 4 , . . . , e } ) . P r o o f . Let C be an s k , L - {0, 1, 2 , . . . , R}), i.e., a perfect code with covering radius R. For any distinct ei and c2 in C, d(ei, e2) > 2R + 1. Now let C ' - C @ IF (cf. Section 3.2). Obviously, for any distinct e~ and e~ in C', d(c~, c~) equals 1 or is at least 2R + 1. Thus L'(e~) and L'(e~) have an empty intersection. Furthermore the sphere-covering equality for C can be rewritten as
2k+iE iEL'
(n+1)-2'~+1 i
hence C' is a perfect L'-covering. Conversely, let C' be an s + 1, k + 1, L') with e - 0 (respectively, 1). Then the 2"~ even vectors in IFT M are covered by the even (respectively, odd) codewords in C'. Deleting the last component in IF'~+i we get an s k, L). []
C o r o l l a r y 19.1.15 The only perfect L-coverings with L C_ [0, n/2] are the classical perfect codes and those with the following parameters: (2R + 2, 2, {R, R - 2 , . . . , R - 2 [R/23 }),
s
2"~ - m, {1}), ~(24, 13, {1, 3}),
derived from binary repetition, Hamming and Golay codes, respectively.
[]
Chapter 19. Heterodox coverings
466
So, either when L is included in [0, n/2] (or, by (19.1.2), in [n/2, n]) or when the code is self-complementary, we know all the p a r a m e t e r s of perfect L-coverings (note t h a t for a given self-complementary code, various sets L are possible). We now study a case when L is not included in [0, n/2], to give a flavour of m e t h o d s t h a t can be used (first note t h a t the construction of T h e o r e m 19.1.14 yields a n / : ( n + 1, 1, L ' - {n, n - 2 , . . . , n - 2[n/2~}) - {0 '~+1, 0'~1} obtained from the trivial perfect code {0'~}). The
case
L -
[0,1] U { n }
Let us first give an easy existence result. Theorem
2}).
Proof.
1 9 . 1 . 1 6 For m > 2, there exists an E ( 2 m - 2 , 2 m - 2 - m , {0, 1, 2 m Shorten a H a m m i n g code of length 2 TM - 1.
[]
On the other hand, as we now prove, for l - 2 there is no such perfect code: Theorem Proof.
19.1.17
There is no nontrivial s
k, {0, 1, 2, n}).
The sphere-covering equality would give, setting n n 2 + n-
k - r"
2(2 ~ - 2) - 0.
The discriminant 2 ~+3 - 15 can be a nonzero square only if 2~ + 3 - z which implies z 2 - 1
2
(mod3),
(mod 3), r + 3 - 27 and 15 - 2 r+3 - z 2 - (2 ~ - z)(2 "r q- z).
Solving for 7 and z gives only the trivial E(3, 0, [0, 3]) code.
[:]
We conclude with a conjecture. C o n j e c t u r e 1 9 . 1 . 1 8 Let L - L1 U L2 U L3 U L4 be the decomposition of L as before. If L i r 0, for i - 1, 2, 3, 4, then L1 U L2 and L3 U L4 are intervals containing 0 and n, respectively. Before giving some results in the nonbinary case, we remind the reader t h a t Corollary 18.2.15 states t h a t no perfect L-covering exists for L - [ ( n w ) / 2 , (n + w)/2].
19.2.
Perfect coverings by spheres of two radii
The
nonbinary
467
case
Here, the situation is simpler and perfect L-coverings are classified as follows. T h e o r e m 19.1.19 For q > 2, the only possible perfect L-coverings occur for L - [0, R], i.e., are the classical ones. The proof follows from two easy lemmas (cf. Lemmas 19.1.5 and 19.1.6). L e m m a 19.1.20 If C is a perfect L-covering, there is no triangle in 7/qn with side lengths t, ~t, d(c, c'), with ~, ~' in L and c, c r in C. [] L e m m a 19.1.21 In 7/q,9t q > 2, there ezists a triangle with side lengths a, b, c if and only if the triangle inequalities are satisfied. [] P r o o f of T h e o r e m 19.1.19 First note that there is no l in L with l > n/2, because a triangle with side lengths ~, ~, w(c) (with c E C), possible by Lemma 19.1.21, would violate Lemma 19.1.20; hence L C [0, n/2]. Suppose that L is not an interval [0, R]. Let x be such that w(x) - min{i" i ~ L} andmmax{i" i E L). T h e n x ~ L(0) and there is a c i n C f o r which x E L(c). Now w(c) < m + w(x) < 2m, so there is a triangle with side lengths m, m, w(c) (by Lemma 19.1.21), which is impossible, by Lemma 19.1.20. []
19.2
Perfect coverings by spheres of two radii
Let us consider another generalization of perfect codes, where the spheres have different radii. Let C - Ul<~
468
Chapter 19. Heterodox coverings
since
[7)*[" V( 2m
-
1, 2 ) + [ ~ , ~ [ - [ ' P * [ - 22'~-1
Setting C2 - 7~*~{0}, R2 - 2, C1 - (7/,~\7~)(9{1}, R1 - 1, it is again easily checked by counting that we obtain a partition of IF 2"~ with the announced parameters. []
T h e o r e m 19.2.2 There exists a P(15, {3, 1}, {32,896}). P r o o f . Take for C1 the [15, 5, 7]5 punctured Reed-Muller code 7~A4"(1, 4) with R1 - 3. There are 28.25 vectors at distance 5 from 7~A4" (1, 4), the socaned "bent" functions (see Notes on Section 9.2), and they form a (15,896, 3) code C2. Set R2 - 1 and conclude by noticing that 3
1
i=0
i=O
-
21s.
T h e o r e m 19.2.3 There exists a P ( l l , {3, 1}, {2, 132}). P r o o f . There exists a Steiner system S(4, 5, 11) (see Example 3.3.3 where it is used to prove that K ( l l , 1) < 192). Together with its complement S, this gives an (11,132, 3) code C1 with the property that for all x 6 IF 11 such that 4 _ w(x) _< 7,-there exists a unique codeword e 6 C1 with d(x, e) _< 1. Take C2 - {011, 111}, R2 - 3 to obtain the existence of P ( l l , {3, 1}, {2,132}). []
T h e o r e m 19.2.4 For s > 4, a P(2s + 3, {s - 1, 1}, {K1, K2}) exists only if g l - 2, K 2 - 2(28~3)/(s + 1). P r o o f . Suppose there exists a P ( 2 s + 3 , { s - l , 1}, {K1, K2}). One can always centre a sphere of radius s - 1 at 0. Suppose K1 >_ 3. Then there would be at least two other spheres Bs_x(x) and Bs-I(y). Now B s _ l ( 0 ) A B s - l ( x ) 0, B s - l ( 0 ) n B , _ l ( y ) - 0, implies w(x) >_ 2 s - 1, w(y) >_ 2 s - 1. For s > 4, B s - l ( x ) and B s - l ( y ) would intersect in 1, the all-one vector of length 2s + 3, which is impossible. Hence K1 _< 2. Suppose now K1 - 1, so the only sphere of radius s - 1 is centred at 0. Call Jr/ the number of spheres whose centres have weight i. Elements of weight s in IF 2~+3 are contained in spheres with centres of weight s + 1. Every such sphere contains s + 1 elements of weight s, so .h,+l - (2s+3)/(s + 1). The same method then shows that A,+2
-
A,+I,
A,+3 - A,+4
-
0.
19.2. Perfect coverings by spheres Of two radii
469
Now spheres with centres of weight s + 5 must contain all elements of weight s + 4 and at most the total number of elements of weight s + 6, hence
~t~+~
_
§
\ ~ - 1 ] / ( ~ + 5)
and
A~+~ <
/ ( ~ - 2) -
+-----~
/(~ +
5),
a contradiction. Finally K1 - 2. Now the sphere-covering equality
K1V(2s + 3, s - 1 ) + K2V(2s + 3, 1 ) - 22'+3 gives K 2 - 2(2"+3] /(s -t- 1). k $ ! Puncturing sometimes allows to obtain new perfect codes from existing ones. Let us mention the following constructions: T h e o r e m 19.2.5 Let P(n, {R1, R2}, {K1, K2}) be such that
K1
()
n-1 R1
-K2
(. 1) R2
;
if moreover any two adjacent spheres of the same radius have the same i-th component, then puncturing on the i-th position yields two perfect 2-radius codes: P(n - 1, {R1 - 1, R2}, { g l , g2}), P(n - 1, {R1, R2 - 1}, {K1, K2}) (two spheres of radius Rj are adjacent if the distance between their centres is
2Rj + 1). Proof.
Straightforward.
E x a m p l e 19.2.6 Viewing the Golay code as a P(23, {3, 3}, {211,211})gives rise to a P(22, {3, 2}, {211, 211}). In fact a further puncturing gives perfect a 3-radius code P(21, {3, 2, 1}, {21~ 211, 21~
[]
Chapter 19.
470
19.3
Heterodox coverings
Coverings by spheres all of different radii
In this section we consider the problem of coverings by spheres when no two equal radii are allowed. Namely, we allow the occurrence in a covering of at most one sphere of each of the radii 1, 2 , . . . , n. The problem is to find a covering minimizing the m a x i m u m radius of a sphere used in the covering. Evidently, one sphere of radius n is enough to cover all the space. As well, two spheres of radii In~21 - 1 and In~21 centred at two complementary vectors suffice. Thus, an evident upperbound on the m a x i m u m radius of a sphere involved in the covering is In~21 . Surprisingly enough we cannot do better. Let us rephrase the problem more precisely. Let I - {R~,..., R~}, R~ < R2 < . . . < Rm, be a subset of { 1 , . . . , n } , m < n and C - { e l , . . . , Cm}, be a collection of centres. Assume that m
[.J
- IF"
j=l
Given n we want to construct a covering C with minimal P ~ . The m i n i m u m of Rm over all possible C is denoted by p(n). L e m m a 19.3.1 Let & - ( a i ) - ( a i , j ) , i - 1 , . . . , n , j 1 , . . . , n , be a matrix with entries 4-1. Then there exists a 4-1-vector y+ - ( y ~ ) such that
I E Y~a',jl<2i l <_j<_,~
for i - - 1 , . . . , n. Proof. For each i, consider the homogenous system of n equations zi = n ~ j = 1 ai,jxj in the n real variables x l , . . . , xn. Set x - ( e l , . . . , xn) and y - ( y l , . . . , yn), where the yi's belong to { - 1 , 0, +1}. A trivial solution to A x T - On is y - On. At step one, relax, say, the last equation: delete the last row of .4, and denote by _4,1 the resulting matrix; then the set of solutions to A1x T - 0n-1 is a vector space of dimension at least one, thus containing a line through the origin. This line hits the real cube [ - I , 1]n on a face; after possible reordering, we can assume that it is the n-th face, of equation xn equal to, s a y , - 1 . Set y - (On - I ~ - 1 ! and repeat this procedure n - I times. T h a t is, before step t + ~ y - ( , , ] y n - t + 1 , . . . , yn) where y n - t + l , . . . , yn are t +1 elements determined by the previous moves; at step t + 1, relax the ( n - t ) - t h equation, determine the space of solutions to the system of n - t - 1 equations in n - t unknowns, find a face where it intersects the cube [ - I , 1]n-t. This fixes a new component of y, say S/n-t, and defines y - ( O n - t - 1 1 y n _ t , . . . , y n ) . Denote the final 4-1 result, obtained after step n, by y+ and consider the sum ~ 1 < j < ~ y~:aij. From the description of the
19.3.
Coverings by spheres all of different radii
471
process, this sum remains 0 until we reach step n - i + 1. During step n - i + 1, the absolute value of this sum increases by at most one, and by at most two during the i - 1 remaining steps. So [ ~ i < j < , ~ Y~ai,Jl < 1 + 2 ( i - 1) < 2i. []
Now we are in position to prove Theorem
19.3.2
,o(,-,)- r,-,/2l. P r o o f . Assume the contrary, i.e., that we have a covering C - { c i , . . . , c,,~}, with R ~ < ~n/2] - 1. W i t h o u t loss of generality, we can assume that R1 - 1, R2- 2,..., Rm-m, with m ~ n / 2 1 - 1 . Let c • be the vector obtained from e by changing O's to l's and l's to - l ' s , and let E be the complement of c. Construct a m a t r i x / k having as first rows the words era_i,• . . . , c~, the • remaining rows being arbitrary. By L e m m a 19.3.1, there is a + l - v e c t o r y such that l_
i - 1 , . . . , m - 1. Since ~l<_j
This problem has an application to broadcasting in the n-dimensional cube. Let 2'~ processors be connected as in the n-dimensional cube, i.e., each processor is assigned a different binary n-tuple and two processors are connected if the corresponding binary vectors are at Hamming distance one. Now, to broadcast a message in the n-cube, it is allowed at each round to provide it to exactly one of the processors; in parallel, a processor that already received the message sends it to its neighbours. Thus, after the first round, one processor, say pi, knows the message; after the second round, pi, its neighbours and some other processor p2 know the message, and so on. The question is: how m a n y rounds are needed to deliver the message to all processors? Theorem 19.3.2 gives the answer, namely In/2] + 1, and the strategy is very easy: just send the message to two antipodal nodes during the first two rounds and after this, ... relax.
Chapter 19. Heterodoz coverings
472
19.4
M u l t i c o v e r i n g radius
In this section we study codes with the property that for every s-tuple of vectors in the Hamming space there is a codeword which is close to all of them. This leads to the concept of multicovering radius. D e f i n i t i o n 19.4.1 Let s >_ 1 be fized.
The multicovering radius of a code C C IFn is the smallest integer R, - Rs(C) such that for every s-tuple (xl, x 2 , . . . , xs), where X1, X 2 , . . . , Xs E IFn, there is a codeword c E C such that d(c, xi) < Rs for all i - 1 , 2 , . . . , s . When s Theorem
1, the definition of course reduces to that of covering radius.
19.4.2 If C is an (n, K) code, then Rs(C) > n - t(n, s).
P r o o f . Recall that t(n, s ) i s the smallest covering radius among all (n, s) codes. Take as the vectors x t, x 2 , . . . , xs the codewords of a binary code of length n with covering radius t(n, s). If y C IF'*, then y is within distance t(n, s) from at least one of these s vectors, say xi, and therefore d(y, xi) >_ n - t(n, s). In particular, any codeword in C is at distance at least n - t(n, s) from at least one xj. []
C o r o l l a r y 19.4.3 If s > 2, then the multicovering radius Rs of any code of
The following theorem shows that determining Rs(IF '~) for all s and n is equivalent to determining the values of K(n, R) for all n and R. Theorem
K(n,n-t-
10.4.4 The quantity Rs (IF'~) equals the smallest integer t such that 1) > s.
P r o o f . We see that R,(IF '~) _< t if and only if for every s-tuple of vectors, say x t , x 2 , . . . , x , there is a point c E IFn such that d(xi, c) ___ t for all i; or equivalently that for every s vectors x t , . . . , x, there is a vector c C IF'~ such that d(xi, 3) >_ n - t. But this is equivalent to saying that K ( n , n - t - 1) > s. []
Determining Rs(IF") is an essential problem also from the point of view of studying the multicovering radius of other codes as can be seen from the following immediate result.
19.5. Perfect coverings of a sphere and constant weight coverings
473
Theorem
19.4.5 For every code C, R,(IF '~) ~ R,(C) ~_ R , ( I F n ) + R ( C ) .
Theorem
19.4.6 R2 (7/m) - 2 m- t.
[]
P r o o f . Let n -- 2' ~ - 1. By Corollary 19.4.3, R2(7/m) >_ r~/21 = 2 ~ - , . it remains to show that R2(7/m) _< (n + 1)/2. Let x, y E IF'~ be arbitrary. If d(x, y) <_ n - 1, then let z C IF '~ be any vector such that d(z, x) _< ( n - 1 ) / 2 and d(z, y) <_ ( n - 1 ) / 2 . Because 7/,,~ has covering radius one, there is a codeword c C 9/,,~ such that d(e, z) __ 1. Then d(c, x) _ (n + 1)/2 and d(c, y) _ (n + 1)/2. Second, assume that d(x, y) = n, i.e., y = ~, and show that there is a codeword c C 7/,~ such that d ( x , c ) E { ( n - 1 ) / 2 , ( n + 1)/2}. Then also d(y, c) = n - d ( x , c) e { ( n - 1)/2, (n + 1)/2}. Let I-I be a parity check matrix of 7/,~ and denote s = t t x T. It is sufficient to show that there is a vector u E IF ~ of weight (n - 1)/2 or (n + 1)/2 such that H u T = s, because then x+uEC. Take I_i_ ( 0 vl
0 v2
... ...
0 v(~-t)/2
1 0
1 vl
1 v2
... ...
1 v(,~-t)/2
)
where the vectors vi run through all the nonzero binary ( m - 1)-tuples. Clearly, the columns on the left hand side add up to 0, and the same is true for the right hand side. If the column s occurs on the right hand side, then all the columns on the right hand side except s add up to s, and we are done. If s appears on the left hand side, then all the columns on the left hand side except s ~aa ~p to ~, b~t now t h r ~ just ( ~ - 3)/2 r ~o w~ have to replace one of them, say (0, vi) T, by the two columns (1, vi) T and (1, 0) T from the right hand side, and we are done. []
19.5
Perfect coverings of a sphere stant weight coverings
and
con-
Consider now the sphere B~(0) of radius r centred at 0 in IF'~. We search for the minimum number of spheres necessary to partition B~(0) into smaller spheres. In other words, we are looking for the minimum number of elements ci C Br(0) which cover B~(0), where each ci covers the sphere Br,(c,) (r, ~ ~ - ~(~)). Consider S~(0), the surface of S , ( 0 ) . Then a partition of Br(0) contains a covering C of St(0), whose elements are the centres of the spheres tangent
Chapter 19. Heterodox coverings
474
to S~(0). Let C~ be the set of elements of C of weight w, r >__ w > 0. Then C~ is a constant weight code and, unless it has only one element, its minimum distance d is even, with d/2 _> r - w + 1 (note also that 2w _ d, i.e., w ___ (r + 1)/2). Using the notation of Section 2.7, we see that [C~ I >__ K(n, w, r, r - w). On the other hand, by Lemma 12.6.8,
IC~o,< n ( n - 1 -
w
w-
(n-w+d/2) d/2
1"'"
) ....
(19.5.1)
Since C~ covers A~o points of S~(O), where
inequality (19.5.1) implies that A~ r ( r - 1 ) . . . ( r - ( r - w) + 1) (~) -< ( n - 2 w + r ) : . . ( n - 2 w + r - ( r - w ) + l ) " Asymptotically, for r fixed" A~ - O(n~). That is, A~ - O(n ~-1) for w-r-landA~-O(n ~-2) for O < w < r - l. So ~ =r-1 l A, - O(n ~- 1) and there are 12(n~) isolated points (spheres of radius zero)in S~(0)N C. Non-asymptotically, supposing that r _~ n/3: A
(:) < ~)
m
~'~?" -- TM
where # - r / ( n - r ) . The sum of the above for 0 < w < r is at most # / ( 1 - # ) . So the proportion of isolated points in a covering of S~(0) is at least 3r
1 - 2#
n-
1-#
n-2r
In other words, C contains at least
(:)~
n - 2r
isolated points. Loosely speaking: If you want to partition a sphere into smaller ones, you have to take a lot of spheres of radius 0 (isolated points). The nontrivial ones cover only a negligible amount of the total volume. Notice that this problem is related to that of perfect p-radius codes dealt with in Section 19.2: namely, by adding the sphere B n - r - l ( 1 ) to a perfect covering of Br(O), we get a perfect covering of IF'~ by spheres with different radii. The following classical theorem in graph theory now solves the problem of covering $3 with elements of $2.
19.6.
Notes
475
T h e o r e m 19.5.2 If the graph G has n vertices and m edges, where m > n2/4, then G contains a triangle. P r o o f . Let G : (V, E) be a graph with no triangle, and let d(u) denote the degree (i.e., the number of adjacent vertices) of a vertex u E V. Then two adjacent vertices have disjoint sets of neighbours, and d ( u ) + d(v) <_ n for every edge {u, v} e E. Trivially ~ v d(v) = 2m and therefore summing over all edges, we get
4m 2
ran>_ Z v6.v
n
vEv
proving our claim.
[:]
On the other hand, a bipartite graph G - (V,E) with V - V1 u V2, E - Vl • V2 shows that there exist graphs with n vertices, [n/2J [n/2] edges and no triangle. Now let C C_ $2 be a 1-covering of $3 in IF'~. We consider the graph G(C) - (V,E) with V - { 1 , 2 , . . . , n } and E - {{i,j} 9e, + e j E C}. This graph has the following property: G(C) contains at least one edge of any triangle T in the complete graph on V. Indeed, otherwise the element of $3 corresponding to T would not be covered by C. In other words, the complementary graph of G(C) is triangle-free, i.e., contains at most n2/4 edges by Theorem 19.5.2. Hence C has size at least ( ~ ) - n2/4 and this lower bound can be achieved"
Ivy1- [Ivl/2J, Iv2l- FlVl/2]
C o r o l l a r y 19.5.3 For all n, K(n, 2, 3, 1 ) - [n(4-2) ~.
More generally, T h e o r e m 19.5.4 If n - t ( r - 1 ) + r0 with 0 < ro < r g ( n , 2, r, r - 2) - (r - 1) (t2) + tro.
19.6
1, r > 2, then [:]
Notes
w L-packings were introduced by Karpovsky [363], where their practical significance - correction or detection of physical failures in networks - is described. In [363] some perfect L-codes are given, as well as an analogue of Lloyd's theorem (cf. Theorem 11.2.1). See also Deza, Karpovsky and Milman
[199].
476
Chapter 19. Heterodoz coverings
Corollaries 19.1.7, 19.1.8 and 19.1.15, Theorems 19.1.9, 19.1.12, 19.1.13 and 19.1.14 are by Cohen and Frankl [148]. Theorem 19.1.16 is due to Karpovsky [363]. Theorem 19.1.17 is by Laurent (unpublished). Reliquet (unpublished) has used methods similar to those of Cohen and Frankl [148] to rule out the existence of nontrivial s k, L) for L - [0, 2t + 1] U {n} and L - {1, 3 , 5 , . . . , 2 t + 1, n} when 3 < 2t + 1 _ n/2. Using earlier results proved by Laurent (unpublished), we get that nontrivial perfect L-coverings with L - [O,l] U {n} can only exist if ~ - 1 or ~>6. The nonbinary case is treated by Cohen and Frankl [149]. w Applications of perfect p-radius codes to source coding and unequal protection of messages are described by Montaron and Cohen [495]. Theorems 19.2.1, 19.2.2, 19.2.3 and 19.2.5 as well as Example 19.2.6 are by Cohen and Montaron [167] [495]. Theorem 19.2.4 is due to Cohen and Frankl [150]. More can be said about Theorem 19.2.1; first, another construction, with the same parameters as, but nonisomorphic to, that of Theorem 19.2.1, was obtained by van den Akker, Koolen and Vaessens [13], by letting C2 7~m, R2 - 2, C1 - ?/m \ 7~,~, R1 - 1, where ^ denotes extension. Now let us call two codewords c E C{, c' E Cj adjacent if d(c, c') R~ + Rj + 1. Then to a code C one can associate in an obvious way a graph F(C) whose set of vertices is C, C being caned bipartite if r ( c ) i s , with C1 and C2 being independent sets (i.e., there are no edges within C1 and C2). For example, the code C - (?/,~ \7~,,~)u7~,,~ is bipartite, since d(c, c') >_ 6, (respectively, 4) if c, c' E C2 (respectively, C~). But the code of Theorem 19.2.1 is not bipartite. Based on the fact that a nearly perfect code which is not perfect and has covering radius at least 3, has the same parameters as a punctured Preparata code (see Theorem 11.6.3), it is proved in [13] that a bipartite 2-radius (R, 1)perfect code with R > 2 necessarily has parameters given by Theorem 19.2.1. On this topic, see also Zinoviev and Katsman [711]. w Lemma 19.3.1 is due to Beck and Spencer [63]. Its use for finding the best strategy for transmitting in the n-cube is by Alon [15]. w The problem discussed in this section is due to Klapper [377] and arose from investigations concerning the cryptanalysis of stream ciphers [376]. Corollary 19.4.3 and Theorem 19.4.6 are from [377], where a more detailed discussion about the multicovering radius and, e.g., its behaviour in various constructions, can be found. w Theorem 19.5.2 is due to Mantel [468]. A result by Turs [656] settles the more general case of covering S~ with a subset of $2. On the
19.6.
No~es
477
subject of perfectly covering a sphere, see Fachini and K;Srner [227] for a proof that no sphere can be partitioned into two spheres, i.e., there is no perfect code in IF'~ with three spheres of any, not necessarily equal, radii. See Etzion, Wei and Zhang [226] for a survey on constant weight coverings.