- Email: [email protected]
On Cliques and Partitions in Hamming Spaces
Annals of Discrete Mathematics 17 (1983) 21 1-217 0North-Holland Publishing Company
ON CLIQUES AND PARTITIONS IN HAMMING SPACES Gerard COHEN ENST, 46 rue Barrault, 75013 Paris, France
Peter FRANKL CNRS. Paris, France Soit H, = (Z/qZ)”, I’ensemble des n-uples d’entiers modulo q, muni de la mttrique de Hamming H: pour x = ( x , ) et y = (yq) dans Hq, H(r, y ) = ({i,xi# y,}]. Une R-clique S, R C 11.2,. . . ,n } ,est un sous-ensemble de H, tel que pour x et y quelconques dans S, H ( x , y ) prend ses valeurs dans R. Soit m ( n , R ) la cardinalit6 maximale d’une telle clique. Pour E C { 1 , 2 , ..., n},uneE-sphitredecentrex e s t E ( x ) = { y E H , , H ( x , y ) E E } . O n a I E ( x ) l = (q - 1)’(;). Un E-code C est un sous-ensemble de H, pour lequel les E-sphires centrtes sur ses 6liments sont d’intersection vide deux i deux. C est parfait si de plus ces E-spheres partitionnent H,.Quand E = {0,1,. . . ,e } , nous parlerons de e-sphere et de e-code. Nous montrons dans cet article 1es rCsultats suivants.
c,,,
Proposition 1. Pour q # 2 les seuls E-codes parfaits possibles sont les e-codes.
Proposition 2. m(n, R ) S inf(C.jR=b(q- 1 ~ ( : ) , qm(n - 1, R ) , ( l / q ) m ( n+ 1, R u R R+l={r+l, rER}. Proposition 3. Pour q = 2, R = {0,2,4,. . . ,2e} et n E-sphire, auec E = { e , e - 2 , e - 4 ,.... e - - 2 [ e / 2 ] } .
3 2e
+ 2,
+ 1))
oli
m(n, R ) est re‘alise‘ par une
Proposition 4. Pour q # 2, R = (0.1,. . . ,2e}, n assez ,grand, m(n, R ) est rialise‘ par une e-sphire.
1. Introduction We consider Hq = ( Z / q Z ) ” , the set of n-tuples of integers mod q, endowed with the Hamming metric H , defined for two elements x = ( x i ) , y = (yi) in H4 as the number of their different coordinates: H ( x , y) = I{i,xi# yi}l. Denote by 0 the all zero element and define the weight of x by w (x) = H ( x , 0). An R-clique S is a subset of Hq s.t. for any x and y in S, H ( x , y) E R C {1,2,. . . ,n}. The maximal cardinality of an R-clique is denoted by m ( n , R). For E C {1,2,. . . , n } , an E-sphere of center x is E ( x ) = {y E H,, H(x, y ) E E } ; obviously, IE(x)I = CICE(q - l)i(?). An E-code C is a subset of H, s.t. for any c I and cz in C, E(cl)n E ( c 2 )= 0. C is perfect if, moreover, { E ( c ) ,c E C} is a partition of H,, which implies 1 E ( x ) / .1 Cl = 9”.In the classical case when E = {0,1,. . . ,e } , we 211
212
G. Cohen. P. Frank1
shall speak of e-spheres and e-codes, which are R -cliques for R = {0,1,. . . ,2e} and R = {2e + 1,2e + 2,. . . , n}, respectively. 2. Inexistence results for E-perfect codes
Proposition 1. For q # 2, the only possible E-perfect codes are e-codes.
The proof will follow from two easy lemmas. Lemma 1. There is no triangle in H, with side lengths e l , e,, H(x, y), with e l , e2 in E and x, y in C. Proof. By translation and permutation of components, this would yield a triangle with vertices c I , c2, z , with cl, c2 in C and z E E ( c l ) n E(c2). Lemma 2. In H, there exists a triangle with sides a, b, c iff the triangle inequalities are satisfied. Proof of Proposition 1. First note that there is no e in E with e > Ln/2J, because for any c I , c, in Hqr H ( c l , c,) S n. So there would exist a triangle with sides e, e, H(c,, c2), violating Lemma 1 - hence E C {0,1,... , Ln/2J}. Suppose (which is always possible by translation) that 0 E C, and E is not an interval {0,1,. . . , e}. Let x be such that w(x) = min i, i e E, and m = max i, i E E. Then x e E(O), and there is a c in C for which x E E(c). Now w ( c ) S m + w ( x ) < 2 m , so there is a triangle with sides m, m, w(c), which is impossible. Remark. The case q = 2, which is more complicated because only a weaker form of Lemma 2 holds, is settled in [2]. If E C [0, Ln/2J], binary E-perfect codes, apart from e-codes, exist iff n = 2", E = {l},or n = 24, E = {1,3},or n = 2e + 2, E ={e,e - 2 , e - 4 ,..., e -2[e/2J}. 3. Bounds for R-cliques
Let us give now upper bounds on m(n, R). Proposition 2. m(n, R)Sinf(A1,A2,A3,A4),where A l = z!?:,(q - 1)'(;), A, = qm(n - 1, R), As = (l/q)m(n + 1, R U { r + 1, r E R}), A 4 = (n, R'), R C R'. Proof. m (n, R ) 6 A I is given by Delsarte [3]. Suppose S is a maximal R -clique.
213
Cliques and partitions in Hamming spaces
Let Si = {x E S, x, = i}. The {Si}, i = 1 , 2 , . . . ,q partition S, so one of them, say So, has cardinality greater than ( 1 l q ) m (n,R ) . Deleting the last component of the elements in So we get SA which is an R-clique in length n - 1, hence m ( n , R ) < q 1 Sol= q I SAl S qm ( n - 1, R ) = A Z .The proof of rn (n,R ) S A 3 is similar. Start from a maximal R-clique S and add an extra component in all possible ways. Then n + n + l , R + R U { r + l , r E R } and I S l + q l S I . r n ( n , R ) S A 4is trivial, We investigate now the following problem. For which E, R , is an E-sphere a maximal R -clique? 3.1. The binary case
In the binary case (q = 2), it is known that for R = {0,1,.. . , 2 e + E } , E = 0, 1, m ( n , R ) is realized by an e-sphere for E = 0 and a quasi ( e + 1)-sphere (cf. Definition 1) for E = 1 (Katona [7]). We now show the following. Proposition 3. For q = 2 , R = { 0 , 2 , 4 , .. . , 2 e } and n 2 2e + 2 , m ( n , R ) is attained by an €-sphere, with E = {e, e - 2, e - 4,. . . , e - 2 [ e / 2 ] } .But for n = 2e or 2e + 1 , m ( n , R ) = 2 " - ' . Proof. Define U,, : H,+ H, by
u,,(XI, xz,. . . , X I , . . . ,x,, . . . ,x,)
= (XhX2,.
. . ,o,.
if x,x, = 1 and U,(x) = x if x,xJ = 0, i.e. w(U,, (x)) also for i < j, V,J(XI,...,X,-r,oIX,+I,XJ
I = (XI,.
. .,o,. . . ,X")
= w(x)
or w(x)-2. Define
1 9 1 7 . . . ? x " )
. .,X8.-1,1, & + I , . . . ,X,-l,O,. . . ,X"),
and V, (x) = x in the other cases (when (1 - x,)x, = 0 or i > j ) . For a maximal R-clique S, set O , , ( S )to be 0, ( S ) = {O,,( v 1; u E s, ot,( u )
e S ) u { u ; u E s,0 ,
(0)E
s>,
where 0,)is U, or V,,. It is obvious that for R = { 0 , 2 , 4 , .. . , 2 e } , O,J( S ) is also an R -clique with of course I O,,( S ) l = IS 1 = m (n, R ) . Iterating for all i, j , one eventually gets an R-clique T stable by the O,J.Now we show that for n 2 2e + 2 , m (n,{ 0 , 2 , .. . , 2 e } )s m ( n - 1, { 0 , 2 , .. . , 2 e } )
+ r n ( n - 1 , { 0 , 2 ,..., 2 e - 2 ) ) . Let T be written T = Tn U TI as in the proof of Proposition 2 .
(1)
G. Cohen, P. Frank1
214
1 TAI = 1 To(s m ( n - 1 , { 0 , 2 , ... , 2 e } ) . We want to show that for any x', y ', in TI, H ( x ' , y ' ) S 2e - 2, or equivalently, for any x, y in T, with x, = y, = 1, H ( x , y ) < 2e. Suppose the contrary. Let x, y be in T , with H ( x , y ) = 2e. As n 5 2e + 2, there exists i, s.t. x, = y i . There are two cases: (i) x, = yi = 1. Then U,n(x) = z E T and H ( z , y ) = 2e + 2 which is impossible. (ii) xi = yi = 0. Then V,,(x) = z and we get also a contradiction, proving (1). Using (1) and
we see that if Proposition 3 holds for some no, it will hold for any n > no.
Lemma 3. Proposition 3 is true for no = 2e
+ 2.
Proof. For R = { 0 , 2 , .. . , 2 e } , it is clear that all elements in an R -clique T have weight of the same parity. Denote by ti the number of elements in T with weight i, and by f the complement of x. One has
w ( 2 )= no - w (x),
H(x, 2 ) = no> 2e,
so x E T j fe T, hence
Suppose e even and all weights in T even (always possible by translation). Then
But this bound is achieved by an E-sphere with E = {0,2,. . . ,e } . The case e odd is similar. Hence the lemma is proved, and also the proposition for n 3 2e + 2.
For n = 2e or 2e R -clique.
+ 1, take
all elements of even (or odd) weight for the
3.2. Quasi -spheres Definition 1. A quasi E-sphere of center x is a set Q E ( c ) such that 3i E E,
( E - { i ) ) ( x ) C Q E ( x )C E ( x ) . When E = {0,1,.
. . ,e } and
i = e, it will be called a quasi e-sphere Q e ( x ) .
Cliques and partitions in Hamming spaces
215
The following extension of [8, 5.2.21 is easy to check.
A quasi E-perfect code will be a partition of H, by QE-spheres. Proposition 1 can be generalized to
Property 2. For q # 2, the only possible QE-perfect codes are Qe-codes. Example. A linear quasi-perfect code (n,K,2e Q ( e + 1)-perfect code.
+ 1) [9,
Chap. 1, $51 is a
By analogy with group theory, we now introduce the following. Definition 2. A t-stabilizator is a subset of H, constituted by elements having zeros in t given components. Of course a t-stabilizator is a {0,1,. . . , n - t}-clique of cardinality q"-'.
3.3. The general case
For any q, let us now prove the following. Proposition 4. ForqZ 2, R = {0,1,. . . ,2e + E } , E = 0,1, m (n, R ) is realized by a n - (2e + 8)-stabilizator for n small, by an e-sphere when E = 0 and n large enough, and by a quasi ( e + 1)-sphereof the following type when E = 1 and n large enough : the union of the e -sphere around 0 and the vectors of weight e + 1 with a '1' in a given position.
-
Proof. The first part ( n small) is proved in [ 5 ] . Consider now the case E = 0, n large. Define, as in the proof of Proposition 3, for i, j, 1 C i C n, 1 S j s q - I , W,, H, H, by WI,(XI,X2,.
*
. ,x,-l,j,x,+,,. . . ,x.)
= (XI,&,
. . . ,xl-l,o,x#+,,. . .
1
X")
and W, (x) = x if x, # j. Starting from a maximal R-clique with R = {0,1,.. . ,2e}, we apply all the W,, to get a stable maximal R-clique T. Now we show that every element in T has weight at most e. For suppose there is an x in T with weight e + 1. Call F ( x )the support of x, i.e., F ( x ) = { i , x , # O } , with I F ( x ) l = w ( x ) . For every V' in T, 1 F(u')\F(u)/S e - 1 . If not, for the element D" coinciding with u ' , except on F ( u ) f l F ( u ' ) where it has zero coordinates, one would have u" E T (obtained
G.Cohen,P. Frank1
216
from u' by applying some W i j ) and H(u, u") = I F(u) A F(u")l= IF ( o ) l + I F(u")I 2 2e + 1 ; which is impossible. Hence
is less than
2 (q
i =O
-ly
(I)
= O(n.>
which is the cardinality of an e-sphere, contradicting the maximality of T. Now let us turn to the case E = 1. It can be proved in the same way as above that w ( x ) S e + 1 for every x E T. Let us set
T* = { x E T, w ( x ) = e + l},
9 = { F ( x ) : xE T * } .
As I T / is maximal we have
On the other hand I T * 1 s 1 91 ( q - l ) e + l , i.e., 9 is a family of (e + 1)-subsets of {l,Z, ...,n} with F n F ' Z 0 for F, F I E 9 and I s 1 3 0 ( n e ) . By Hilton and Milner [ 6 ] we deduce 3i such that i E F for all F in 9.By symmetry we may assume i = n. Thus we deduce
which yields
From the proof it is clear that the only way to have equality is by the quasi-sphere. Proposition 4 settles a conjecture of [5] in the case n % 2e + E = n - 1. Namely that m (n, {0,1,. . . , 2 e + E } ) is obtained by a direct product of stabilizators and spheres. 3.4. An example
Propositions 2 , 3 and 4 can be combined to give bounds on m (n, R ). Delsarte's bound m ( n , R ) S A l is good when R is sparse, whereas m ( n , R ) S A , becomes
Cliques and partitions in Hamming spaces
217
betterwhen R isdense(e.g., r m + M a x , ( r , , , - r , ) S 2 1 R I f o r R = { r l , r 2 ,..., m,} with rl < r2 < . . . < r,,,). Let us consider m (6, (0, 1 ,2,4}), q = 2.
rn (6, {0,1,2,4}) L rn (6, {0,2,4}) = 16 (Proposition 3). rn(6, {0,1,2,4}) s irn (7, {0,1,2,3,4,5}) = A,
(Proposition 2),
with A , = 22 (realized by a 03-sphere). Also
rn (6, {0,1,2,4}) s m (6, (0, 1,2,3,4}) = 22 (sphere of radius 2), so 16 6 rn (6, {0,1,2,4}) S 22.
Rosenberg notes that 16 is the exact value. References C. Berge, Nombres de coloration de I’hypergraphe h-parti complet, Hypergraph Seminar, Colombus, Ohio, 1972 (Springer, Berlin, 1974) 13-20. PI G. Cohen and P. Frankl, On tilings of the binary vector space, Discrete Math. 31 (1980) 271-277. [31 P. Delsarte. Four fundamental parameters of a code and their combinatorial significance. Inform. Control 23 (1973) 407-438. [4] M. Deza and P. Frankl, Every large set of (0, + 1, - 1)-vectors forms a sunflower, Combin. (to appear). 151 P. Frankl and Z. Furedi, The Erdos-Ko-Rado theorem for integer sequences,,SIAMJ. Discrete Appl. Methods 1 (1980). [ h ] A.J.W. Hilton and E.C. Milner, Some intersection theorems for systems of finite sets, Quart. J . Math. Oxford Ser. (2) 18 (1967) 369-384. (71 G. Katona, Intersection theorems for systems of finite sets, Acta Math. Acad. Sci. Hungar. 15 (1964) 329-337. [8]J. van Lint, Coding Theory (Springer, Berlin, 1973) 201. 191 F. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam, 1977).
ON CLIQUES AND PARTITIONS IN HAMMING SPACES Gerard COHEN ENST, 46 rue Barrault, 75013 Paris, France
Peter FRANKL CNRS. Paris, France Soit H, = (Z/qZ)”, I’ensemble des n-uples d’entiers modulo q, muni de la mttrique de Hamming H: pour x = ( x , ) et y = (yq) dans Hq, H(r, y ) = ({i,xi# y,}]. Une R-clique S, R C 11.2,. . . ,n } ,est un sous-ensemble de H, tel que pour x et y quelconques dans S, H ( x , y ) prend ses valeurs dans R. Soit m ( n , R ) la cardinalit6 maximale d’une telle clique. Pour E C { 1 , 2 , ..., n},uneE-sphitredecentrex e s t E ( x ) = { y E H , , H ( x , y ) E E } . O n a I E ( x ) l = (q - 1)’(;). Un E-code C est un sous-ensemble de H, pour lequel les E-sphires centrtes sur ses 6liments sont d’intersection vide deux i deux. C est parfait si de plus ces E-spheres partitionnent H,.Quand E = {0,1,. . . ,e } , nous parlerons de e-sphere et de e-code. Nous montrons dans cet article 1es rCsultats suivants.
c,,,
Proposition 1. Pour q # 2 les seuls E-codes parfaits possibles sont les e-codes.
Proposition 2. m(n, R ) S inf(C.jR=b(q- 1 ~ ( : ) , qm(n - 1, R ) , ( l / q ) m ( n+ 1, R u R R+l={r+l, rER}. Proposition 3. Pour q = 2, R = {0,2,4,. . . ,2e} et n E-sphire, auec E = { e , e - 2 , e - 4 ,.... e - - 2 [ e / 2 ] } .
3 2e
+ 2,
+ 1))
oli
m(n, R ) est re‘alise‘ par une
Proposition 4. Pour q # 2, R = (0.1,. . . ,2e}, n assez ,grand, m(n, R ) est rialise‘ par une e-sphire.
1. Introduction We consider Hq = ( Z / q Z ) ” , the set of n-tuples of integers mod q, endowed with the Hamming metric H , defined for two elements x = ( x i ) , y = (yi) in H4 as the number of their different coordinates: H ( x , y) = I{i,xi# yi}l. Denote by 0 the all zero element and define the weight of x by w (x) = H ( x , 0). An R-clique S is a subset of Hq s.t. for any x and y in S, H ( x , y) E R C {1,2,. . . ,n}. The maximal cardinality of an R-clique is denoted by m ( n , R). For E C {1,2,. . . , n } , an E-sphere of center x is E ( x ) = {y E H,, H(x, y ) E E } ; obviously, IE(x)I = CICE(q - l)i(?). An E-code C is a subset of H, s.t. for any c I and cz in C, E(cl)n E ( c 2 )= 0. C is perfect if, moreover, { E ( c ) ,c E C} is a partition of H,, which implies 1 E ( x ) / .1 Cl = 9”.In the classical case when E = {0,1,. . . ,e } , we 211
212
G. Cohen. P. Frank1
shall speak of e-spheres and e-codes, which are R -cliques for R = {0,1,. . . ,2e} and R = {2e + 1,2e + 2,. . . , n}, respectively. 2. Inexistence results for E-perfect codes
Proposition 1. For q # 2, the only possible E-perfect codes are e-codes.
The proof will follow from two easy lemmas. Lemma 1. There is no triangle in H, with side lengths e l , e,, H(x, y), with e l , e2 in E and x, y in C. Proof. By translation and permutation of components, this would yield a triangle with vertices c I , c2, z , with cl, c2 in C and z E E ( c l ) n E(c2). Lemma 2. In H, there exists a triangle with sides a, b, c iff the triangle inequalities are satisfied. Proof of Proposition 1. First note that there is no e in E with e > Ln/2J, because for any c I , c, in Hqr H ( c l , c,) S n. So there would exist a triangle with sides e, e, H(c,, c2), violating Lemma 1 - hence E C {0,1,... , Ln/2J}. Suppose (which is always possible by translation) that 0 E C, and E is not an interval {0,1,. . . , e}. Let x be such that w(x) = min i, i e E, and m = max i, i E E. Then x e E(O), and there is a c in C for which x E E(c). Now w ( c ) S m + w ( x ) < 2 m , so there is a triangle with sides m, m, w(c), which is impossible. Remark. The case q = 2, which is more complicated because only a weaker form of Lemma 2 holds, is settled in [2]. If E C [0, Ln/2J], binary E-perfect codes, apart from e-codes, exist iff n = 2", E = {l},or n = 24, E = {1,3},or n = 2e + 2, E ={e,e - 2 , e - 4 ,..., e -2[e/2J}. 3. Bounds for R-cliques
Let us give now upper bounds on m(n, R). Proposition 2. m(n, R)Sinf(A1,A2,A3,A4),where A l = z!?:,(q - 1)'(;), A, = qm(n - 1, R), As = (l/q)m(n + 1, R U { r + 1, r E R}), A 4 = (n, R'), R C R'. Proof. m (n, R ) 6 A I is given by Delsarte [3]. Suppose S is a maximal R -clique.
213
Cliques and partitions in Hamming spaces
Let Si = {x E S, x, = i}. The {Si}, i = 1 , 2 , . . . ,q partition S, so one of them, say So, has cardinality greater than ( 1 l q ) m (n,R ) . Deleting the last component of the elements in So we get SA which is an R-clique in length n - 1, hence m ( n , R ) < q 1 Sol= q I SAl S qm ( n - 1, R ) = A Z .The proof of rn (n,R ) S A 3 is similar. Start from a maximal R-clique S and add an extra component in all possible ways. Then n + n + l , R + R U { r + l , r E R } and I S l + q l S I . r n ( n , R ) S A 4is trivial, We investigate now the following problem. For which E, R , is an E-sphere a maximal R -clique? 3.1. The binary case
In the binary case (q = 2), it is known that for R = {0,1,.. . , 2 e + E } , E = 0, 1, m ( n , R ) is realized by an e-sphere for E = 0 and a quasi ( e + 1)-sphere (cf. Definition 1) for E = 1 (Katona [7]). We now show the following. Proposition 3. For q = 2 , R = { 0 , 2 , 4 , .. . , 2 e } and n 2 2e + 2 , m ( n , R ) is attained by an €-sphere, with E = {e, e - 2, e - 4,. . . , e - 2 [ e / 2 ] } .But for n = 2e or 2e + 1 , m ( n , R ) = 2 " - ' . Proof. Define U,, : H,+ H, by
u,,(XI, xz,. . . , X I , . . . ,x,, . . . ,x,)
= (XhX2,.
. . ,o,.
if x,x, = 1 and U,(x) = x if x,xJ = 0, i.e. w(U,, (x)) also for i < j, V,J(XI,...,X,-r,oIX,+I,XJ
I = (XI,.
. .,o,. . . ,X")
= w(x)
or w(x)-2. Define
1 9 1 7 . . . ? x " )
. .,X8.-1,1, & + I , . . . ,X,-l,O,. . . ,X"),
and V, (x) = x in the other cases (when (1 - x,)x, = 0 or i > j ) . For a maximal R-clique S, set O , , ( S )to be 0, ( S ) = {O,,( v 1; u E s, ot,( u )
e S ) u { u ; u E s,0 ,
(0)E
s>,
where 0,)is U, or V,,. It is obvious that for R = { 0 , 2 , 4 , .. . , 2 e } , O,J( S ) is also an R -clique with of course I O,,( S ) l = IS 1 = m (n, R ) . Iterating for all i, j , one eventually gets an R-clique T stable by the O,J.Now we show that for n 2 2e + 2 , m (n,{ 0 , 2 , .. . , 2 e } )s m ( n - 1, { 0 , 2 , .. . , 2 e } )
+ r n ( n - 1 , { 0 , 2 ,..., 2 e - 2 ) ) . Let T be written T = Tn U TI as in the proof of Proposition 2 .
(1)
G. Cohen, P. Frank1
214
1 TAI = 1 To(s m ( n - 1 , { 0 , 2 , ... , 2 e } ) . We want to show that for any x', y ', in TI, H ( x ' , y ' ) S 2e - 2, or equivalently, for any x, y in T, with x, = y, = 1, H ( x , y ) < 2e. Suppose the contrary. Let x, y be in T , with H ( x , y ) = 2e. As n 5 2e + 2, there exists i, s.t. x, = y i . There are two cases: (i) x, = yi = 1. Then U,n(x) = z E T and H ( z , y ) = 2e + 2 which is impossible. (ii) xi = yi = 0. Then V,,(x) = z and we get also a contradiction, proving (1). Using (1) and
we see that if Proposition 3 holds for some no, it will hold for any n > no.
Lemma 3. Proposition 3 is true for no = 2e
+ 2.
Proof. For R = { 0 , 2 , .. . , 2 e } , it is clear that all elements in an R -clique T have weight of the same parity. Denote by ti the number of elements in T with weight i, and by f the complement of x. One has
w ( 2 )= no - w (x),
H(x, 2 ) = no> 2e,
so x E T j fe T, hence
Suppose e even and all weights in T even (always possible by translation). Then
But this bound is achieved by an E-sphere with E = {0,2,. . . ,e } . The case e odd is similar. Hence the lemma is proved, and also the proposition for n 3 2e + 2.
For n = 2e or 2e R -clique.
+ 1, take
all elements of even (or odd) weight for the
3.2. Quasi -spheres Definition 1. A quasi E-sphere of center x is a set Q E ( c ) such that 3i E E,
( E - { i ) ) ( x ) C Q E ( x )C E ( x ) . When E = {0,1,.
. . ,e } and
i = e, it will be called a quasi e-sphere Q e ( x ) .
Cliques and partitions in Hamming spaces
215
The following extension of [8, 5.2.21 is easy to check.
A quasi E-perfect code will be a partition of H, by QE-spheres. Proposition 1 can be generalized to
Property 2. For q # 2, the only possible QE-perfect codes are Qe-codes. Example. A linear quasi-perfect code (n,K,2e Q ( e + 1)-perfect code.
+ 1) [9,
Chap. 1, $51 is a
By analogy with group theory, we now introduce the following. Definition 2. A t-stabilizator is a subset of H, constituted by elements having zeros in t given components. Of course a t-stabilizator is a {0,1,. . . , n - t}-clique of cardinality q"-'.
3.3. The general case
For any q, let us now prove the following. Proposition 4. ForqZ 2, R = {0,1,. . . ,2e + E } , E = 0,1, m (n, R ) is realized by a n - (2e + 8)-stabilizator for n small, by an e-sphere when E = 0 and n large enough, and by a quasi ( e + 1)-sphereof the following type when E = 1 and n large enough : the union of the e -sphere around 0 and the vectors of weight e + 1 with a '1' in a given position.
-
Proof. The first part ( n small) is proved in [ 5 ] . Consider now the case E = 0, n large. Define, as in the proof of Proposition 3, for i, j, 1 C i C n, 1 S j s q - I , W,, H, H, by WI,(XI,X2,.
*
. ,x,-l,j,x,+,,. . . ,x.)
= (XI,&,
. . . ,xl-l,o,x#+,,. . .
1
X")
and W, (x) = x if x, # j. Starting from a maximal R-clique with R = {0,1,.. . ,2e}, we apply all the W,, to get a stable maximal R-clique T. Now we show that every element in T has weight at most e. For suppose there is an x in T with weight e + 1. Call F ( x )the support of x, i.e., F ( x ) = { i , x , # O } , with I F ( x ) l = w ( x ) . For every V' in T, 1 F(u')\F(u)/S e - 1 . If not, for the element D" coinciding with u ' , except on F ( u ) f l F ( u ' ) where it has zero coordinates, one would have u" E T (obtained
G.Cohen,P. Frank1
216
from u' by applying some W i j ) and H(u, u") = I F(u) A F(u")l= IF ( o ) l + I F(u")I 2 2e + 1 ; which is impossible. Hence
is less than
2 (q
i =O
-ly
(I)
= O(n.>
which is the cardinality of an e-sphere, contradicting the maximality of T. Now let us turn to the case E = 1. It can be proved in the same way as above that w ( x ) S e + 1 for every x E T. Let us set
T* = { x E T, w ( x ) = e + l},
9 = { F ( x ) : xE T * } .
As I T / is maximal we have
On the other hand I T * 1 s 1 91 ( q - l ) e + l , i.e., 9 is a family of (e + 1)-subsets of {l,Z, ...,n} with F n F ' Z 0 for F, F I E 9 and I s 1 3 0 ( n e ) . By Hilton and Milner [ 6 ] we deduce 3i such that i E F for all F in 9.By symmetry we may assume i = n. Thus we deduce
which yields
From the proof it is clear that the only way to have equality is by the quasi-sphere. Proposition 4 settles a conjecture of [5] in the case n % 2e + E = n - 1. Namely that m (n, {0,1,. . . , 2 e + E } ) is obtained by a direct product of stabilizators and spheres. 3.4. An example
Propositions 2 , 3 and 4 can be combined to give bounds on m (n, R ). Delsarte's bound m ( n , R ) S A l is good when R is sparse, whereas m ( n , R ) S A , becomes
Cliques and partitions in Hamming spaces
217
betterwhen R isdense(e.g., r m + M a x , ( r , , , - r , ) S 2 1 R I f o r R = { r l , r 2 ,..., m,} with rl < r2 < . . . < r,,,). Let us consider m (6, (0, 1 ,2,4}), q = 2.
rn (6, {0,1,2,4}) L rn (6, {0,2,4}) = 16 (Proposition 3). rn(6, {0,1,2,4}) s irn (7, {0,1,2,3,4,5}) = A,
(Proposition 2),
with A , = 22 (realized by a 03-sphere). Also
rn (6, {0,1,2,4}) s m (6, (0, 1,2,3,4}) = 22 (sphere of radius 2), so 16 6 rn (6, {0,1,2,4}) S 22.
Rosenberg notes that 16 is the exact value. References C. Berge, Nombres de coloration de I’hypergraphe h-parti complet, Hypergraph Seminar, Colombus, Ohio, 1972 (Springer, Berlin, 1974) 13-20. PI G. Cohen and P. Frankl, On tilings of the binary vector space, Discrete Math. 31 (1980) 271-277. [31 P. Delsarte. Four fundamental parameters of a code and their combinatorial significance. Inform. Control 23 (1973) 407-438. [4] M. Deza and P. Frankl, Every large set of (0, + 1, - 1)-vectors forms a sunflower, Combin. (to appear). 151 P. Frankl and Z. Furedi, The Erdos-Ko-Rado theorem for integer sequences,,SIAMJ. Discrete Appl. Methods 1 (1980). [ h ] A.J.W. Hilton and E.C. Milner, Some intersection theorems for systems of finite sets, Quart. J . Math. Oxford Ser. (2) 18 (1967) 369-384. (71 G. Katona, Intersection theorems for systems of finite sets, Acta Math. Acad. Sci. Hungar. 15 (1964) 329-337. [8]J. van Lint, Coding Theory (Springer, Berlin, 1973) 201. 191 F. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam, 1977).