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The Diametric Theorem in Hamming Spaces—Optimal Anticodes
ADVANCES IN APPLIED MATHEMATICS ARTICLE NO.
20, 429]449 Ž1998.
AM980588
The Diametric Theorem in Hamming Spaces}Optimal Anticodes Rudolf Ahlswede and Levon H. Khachatrian Mathematik, Uni¨ ersitaat Bielefeld, Bielefeld 33501, Germany Received July 1, 1995; accepted April 21, 1997
For a Hamming space Ž Xan , d H ., the set of n-length words over the alphabet Xa s 0, 1, . . . , a y 14 endowed with the distance d H , which for two words x n s Ž x 1 , . . . , x n ., y n s Ž y 1 , . . . , yn . g Xan counts the number of different components, we determine the maximal cardinality of subsets with a prescribed diameter d or, in another language, anticodes with distance d. We refer to the result as the diametric theorem. In a sense anticodes are dual to codes, which have a prescribed lower bound on the pairwise distance. It is a hopeless task to determine their maximal sizes exactly. We find it remarkable that the diametric theorem Žfor arbitrary a . can be derived from our recent complete intersection theorem, which can be viewed as a diametric theorem Žfor a s 2. in the restricted case, where all n-length words considered have exactly k ones. Q 1998 Academic Press
1. PREVIOUS RESULTS, CONJECTURES, AND THE NEW THEOREM This paper is another demonstration of the power of the methods of w3x. We stick to the earlier notation as far as possible and first repeat it. Then we state the complete intersection theorem in its historical context, because this enables us to put the new result into proper perspective. Here we need some more terminology for the formulation of known results and conjectures for the diametric problem in Hamming space or related intersection problems. Finally, we state the new diametric theorem. N denotes the set of positive integers and for i, j g N, i - j, the set i, i q 1, . . . , j4 is abbreviated as w i, j x. Moreover, for w1, j x we also write w j x. 429 0196-8858r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
430
AHLSWEDE AND KHACHATRIAN
For k, n g N, k F n, we set 2 w nx s F : F ; w 1, n x 4
žw x/ n k
and
s F g 2 w nx : < F < s k 4 .
A system of sets A ; 2 w nx is called t-intersecting, if < A1 l A 2 < G t
for all A1 , A 2 g A ,
Ž 1.1.
and I Ž n, t . denotes the set of all such systems. Moreover, we define I Ž n, k, t . s A g I Ž n, t .: A ; Ž w kn x .4 . The investigation of the function M Ž n, k, t . s max A g IŽ n, k, t . < A <, 1 F t F k F n, and the structure of maximal systems was initiated by Erdos, ˝ Ko, and Rado w5x. THEOREM EKR w5x.
For 1 F t F k and n G n 0 Ž k, t . Ž suitable., M Ž n, k, t . s
ž
nyt . kyt
/
Clearly, the system
w nx
½ ž /
A Ž n, k, t . s A g
k
: w 1, t x ; A
5
y t. Ž . is t-intersecting, has cardinality Ž nk y t , and is therefore optimal for n G n 0 k, t .
The smallest n 0 Ž k, t ., for which this is the case, has been determined by Frankl w6x for t G 15 and subsequently by Wilson w12x for all t: n0 Ž k , t . s Ž k y t q 1. Ž t q 1. . In the recent paper w3x we have settled all the remaining cases: n Ž k y t q 1.Ž t q 1.. The following result plays a key role in the present paper. COMPLETE INTERSECTION THEOREM AK w3x. Define Fi s F g Ž w kn x .: < F l w1, t q 2 i x< G t q i4 for 0 F i F Ž n y t .r2. For 1 F t F k F n with
Ž i. Ž k y t q 1. 2 q
ž
ty1 rq1
/
- n - Ž k y t q 1. 2 q
ž
ty1 r
/
for some r g N j 0 4
431
DIAMETRIC THEOREM IN HAMMING SPACES
we ha¨ e M Ž n, k, t . s < Fr < and Fr is}up to permutations}the unique optimum. By con¨ ention, Ž t y 1.rr s ` for r s 0.
Ž ii .
ty1
Ž k y t q 1. 2 q
ž
rq1
/
sn
for r g N j 0 4
we ha¨ e M Ž n, k, t . s < Fr < s < Frq1 < and an optimal system equals}up to permutations}either Fr or Frq1. Erdos, ˝ Ko, and Rado also initiated the study of optimal systems in I Ž n, t . and of the function M Ž n, t . s
max < A < .
AgI Ž n , t .
A complete description was given by Katona, who, in particular, obtained THEOREM Ka w8x.
¡
M Ž n, t . s
~
n
Ý
is Ž nqt .r2
n , i
n
¢
if n q t is e¨ en,
ž /
Ý
is Ž nqtq1 .r2
n q i
ž /
ny1 nqty1 2
0
,
if n q t is odd.
His proof proceeds by estimating ‘‘shadows’’ of sets in Ž w kn x .. Actually, it also can be proved by the method of the present paper. Now we make a transition from 2 w nx to X 2n s 0, 14 n and the more general Xan s 0, 1, . . . , a y 14 n. Clearly, any set A g 2 w nx can be represented as word a n s Ž a1 , . . . , a n . g X 2n, where ai s
½
1, 0,
if i g A, if i g w n x _ A,
and conversely. Furthermore, in Xan we have a second concept of intersection. We call A ; Xan t y Xa-intersecting, if, for all a n, b n g A, int Ž a n , b n . J j g w n x : a j s bj 4 G t. Let Ja Ž n, t . denote the set of all such systems.
Ž 1.2.
432
AHLSWEDE AND KHACHATRIAN
Since d H Ž a n, b n . s Ž j g w1, n x: a j / bj 4. s n y intŽ a n, b n ., we can equivalently say that A has a diameter diam Ž A . J
max d H Ž a n , b n . F d s n y t
a n , b ng A
or that A is d-diametric. It is important to notice that the notions t y X 2-intersecting in X 2n and t-intersecting in 2 w nx are quite different! We are concerned here with the function Na Ž n, t . s
max
AgJa Ž n , t .
< A <.
Ž 1.3.
There is already a well-known result for a s 2 due to Kleitman: THEOREM K1 w9x.
¡
Ž nyt .r2
N2 Ž n, t . s
~
¢2
n , i
ž / Ý ž
Ý
is0
Ž nyty1 .r2 is0
if n y t is e¨ en, ny1 , i
/
if n y t is odd.
This result and Theorem Ka imply N2 Ž n, t . s M Ž n, t . .
Ž 1.4.
Actually, it was shown in w2x that the two theorems can be easily derived from each other by passing through upsets. Finally, we report on the results known Žto us., which were obtained during the last three decades on Na Ž n, t . for a ) 2. Berge w4x proved that Na Ž n, 1 . s a ny 1
for a G 3.
Ž 1.5.
Livingston w10x showed that the A g Ja Ž n, 1. satisfying Ž1.5. are of the form A s a n s Ž a1 , . . . , a n . g Xan : a i s a4 w7x conjecfor some i g w n x and a g 0, 1, . . . , a y 14 . Frankl and Furedi ¨ tured that Na Ž n, t . s a ny t
iff n F t q 1 or a G t q 1,
Ž 1.6.
and they proved this for t G 15. Ahlswede, et al. w1x remarked that, for
433
DIAMETRIC THEOREM IN HAMMING SPACES
n F t q 1 q log trlogŽ a y 1.,
¡
Ž nyt .r2
Na Ž n, t . s
~
¢a
Ý
n i
i
ž /Ža . Ý ž /Ža y1 ,
is0 Ž nyty1 .r2
ny1 i
is0
if n y t is even,
Ž 1.7. i
y 1. ,
if n y t is odd.
w7x Žand in a diametric formulation also Ahlswede et al. Frankl and Furedi ¨ w1x. have made the General Conjecture. Na Ž n, t . s
max
0FiF Ž nyt .r2
< Ki <
for all n, a , t ,
Ž 1.8.
where, with the convention B Ž a n . s j: a j s a y 14 , Ki s a n g Xan : B Ž a n . l w 1, t q 2 i x G t q i 4 .
Ž 1.9.
Clearly, Ki g Ja Ž n, t . for 0 F i F Ž n y t .r2. We note that in this terminology the results Ž1.6. and Ž1.7. can be summarized in the form
¡< K < , N Ž n, t . s~ ¢< K
if n F t q 1 or a G t q 1, t G 15, log t if n F t q 1 q . log Ž a y 1 .
0
a
?Ž nyt .r2 @
<,
This covers only very few values of the parameters n, a , t. We settle here all cases by establishing the general conjecture. DIAMETRIC THEOREM. such that
For a G 2 let r g 04 j N be the largest integer
½
t q 2 r - min n q 1, t q 2
ty1
ay2
5
.
Then Na Ž n, t . s < Kr <. Ž By con¨ ention, Ž t y 1.rŽ s y 2. s ` for a s 2.. Remark 1. Actually, we also can prove that, up to permutation of 1, 2, . . . , n4 and permutations of the alphabet in the components, there is exactly one optimal configuration, unless t ) 1, t q 2Ž t y 1.rŽ a y 2. F n and Ž t y 1.rŽ a y 2. is integral, in which case we have two optimal configurations, KŽ ty1.rŽ ay2. and KŽ ty1.rŽ ay2.y1.
434
AHLSWEDE AND KHACHATRIAN
Remark 2. A generalization of Theorem Ka to every a G 2 can be obtained with the intersection concept based on the quantity intU Ž a n , b n . J j g w n x : a j s bj s a y 1 4 instead of the quantity intŽ a n, b n . defined in Ž1.2.. In analogy to Na Ž n, t . we get now Ma Ž n, t ., where M2 Ž n, t . equals the familiar M Ž n, t . and obviously ?Ma Ž n, t . F Na Ž n, t .. The structure of the Ki ’s and the diametric theorem imply now Ma Ž n, t . s Na Ž n, t .. In particular, we have thus shown that Theorem Ka can also be proved by our methods.
2. REDUCTION TO CANONICAL AND STABLE SETS We give combinatorial characterizations of Na Ž n, t . of increasing precision. The first one is not new. PROPOSITION FF w7x. n
Na Ž n, t . s
max
Ý < Gi < Ž a y 1. ny i ,
Ž 2.1.
GgI Ž n , t . is0
where Gi s G l Ž w ni x .. We derive this result, because we want to start from first principles and at the same time introduce some concepts. Here it is more convenient to write A for a word a n s Ž a1 , . . . , a n .. It seems that the following transformation was first used by Kleitman w9x. For any A ; Xan, any A s Ž a1 , a2 , . . . , a n . g A, and 1 F j F n, 0 F i F a y 1, we define
¡Ž a , . . . , a , a y 1, a , . . . , a . , this is not element of A and a s i , ¢A, ifotherwise
T Ž A . s~ ji
1
jy1
jq1
n
j
and Tji Ž A . s Tji Ž A . : A g A 4 . Repeated application of these transformations yields after finitely many steps an AX ; Xan, for which Tji Ž AX . s AX
for all 1 F j F n, 0 F i F a y 1.
Ž 2.2.
435
DIAMETRIC THEOREM IN HAMMING SPACES
DEFINITION 2.1. A set A ; Xan is said to be canonical, if for all 1 F j F n, 0 F i F a y 1.
Ti j Ž A . s A
The transformation Tji has the important properties to keep the cardinality and the t y Xa-intersection property unchanged; that is, < Tji Ž A .< s < A < and A g Ja Ž n, t . implies Tji Ž A . g Ja Ž n, t .. Hence Na Ž n, t . s
max
AgJa Ž n , t .
< A< s
max
AgCJa Ž n , t .
< A <,
Ž 2.3.
where CJa Ž n, t . ; Ja Ž n, t . is the set of all canonical systems in Ja Ž n, t .. Now we make the transition to 2 w nx. DEFINITION 2.2. To a system A g CJa Ž n, t . we associate the set-theoretical image BŽ A . s B Ž A.: A g A 4 , where B Ž A. is defined in Ž1.8.. We have an immediate consequence w7x.. LEMMA 1 ŽFrankl and Furedi ¨
For A g CJa Ž n, t .,
B Ž A . g I Ž n, t . . DEFINITION 2.3. For any D g 2 w nx, we define the upset U Ž D . s DX g 2 : D ; DX 4 . More generally Žwith slight abuse of notation. for C ; 2 w nx, we define the upset w nx
U Ž D. s
D U Ž D. .
Dg D
Again, we have a direct consequence of the definitions. LEMMA 2. Let A g CJa Ž n, t . satisfy < A < s Na Ž n, t ., and let BŽ A . be the set-theoretical image of A. Then Ži.
Ž ii .
BŽ A . is an upset. < A< s
Ý
< Ž A .<
Ž a y 1 . ny B
B Ž A .g BŽ A . n
s
Ý g i Ž a y 1. ny i ,
Ž 2.4.
is0
where g i s BŽ A . l
žw x/ n i
.
This yields the proposition. We introduce another familiar concept.
436
AHLSWEDE AND KHACHATRIAN
DEFINITION 2.4. A set of subsets D ; 2 w nx is said to be left-compressed or stable, if, for every D g D and every 1 F i - j F n with i f D, j g D necessarily, DX s ŽŽ D _ j4. j i4. g D. We denote by LI Ž n, t . ; I Ž n, t . the set of all stable systems in I Ž n, t . and by LCJa Ž n, t . ; CJa Ž n, t . the set of all systems A g CJa Ž n, t . with BŽ A . g LI Ž n, t .. From Ž2.2. and the left-pushing technique of w5x, it readily follows that Na Ž n, t . s
max
AgJa Ž n , t .
< A< s
max
AgCJa Ž n , t .
< A< s
max
AgLCJa Ž n , t .
< A <.
Ž 2.5.
Next, for an E g 2 w nx we introduce V Ž E . s A g Xan : B Ž A . g U Ž E . 4 .
Ž 2.6.
V Ž E . s a ny < E < .
Ž 2.7.
Clearly,
More generally, for E ; 2 w nx we introduce V Ž E. s
D V Ž E. .
Eg E
DEFINITION 2.5. Let A g CJa Ž n, t . and let BŽ A . be the set-theoretical image of A. Then A is called a-upset, if AsV
Ž BŽ A . . .
DEFINITION 2.6. For E s e1 , e2 , . . . , e < E < 4 ; w n x, e1 - e2 - ??? - e < E < , write the biggest element e < E < as sqŽ E .. Also for E ; 2 w nx set sq Ž E . s max sq Ž E . . Eg E
The next important properties immediately follow from left-compressedness arguments Žsimilar to those in w3x.. LEMMA 3. Let A g LCJa Ž n, t ., let A be an a-upset, and let BŽ A . be the set-theoretical image of A. Further, let M Ž A . be the set of minimal elements of BŽ A . Ž in the sense of set-theoretical inclusion.. Then A is a disjoint union As
D
EgM Ž A .
DŽ E. ,
437
DIAMETRIC THEOREM IN HAMMING SPACES
where D Ž E . s A s Ž a1 , . . . , a n . g Xan : B Ž A . l 1, sq Ž E . s E and Ž a sq Ž E .q1 , . . . , a n . g Xany s
q
Ž E.
4.
Ž 2.8.
LEMMA 4. For an a-upset A g LCJa Ž n, t ., choose E g M Ž A . such that sqŽ E . s sqŽ M Ž A .. and consider the set of elements of A, which are only generated by E, that is, AE s V Ž E . _ V Ž M Ž A . _ E . . Then AE s D Ž E .
Ž D Ž E . is defined in Ž 2.8. .
and < AE < s Ž a y 1 . s
qŽ
E .y < E <
? a nys
q
Ž E.
.
Ž 2.9.
LEMMA 5. Let A g LCJa Ž n, t ., let A be an a-upset, and let E1 , E2 g M Ž A . ha¨ e the properties i f E1 j E2 , j g E1 l E2 for some i, j g w n x with i - j. Then < E1 l E2 < G t q 1.
3. THE TWO MAIN AUXILIARY RESULTS LEMMA 6. For a ) 2 let A g LCJa Ž n, t . with < A < s Na Ž n, t ., let BŽ A . be the set-theoretical image of A, and let M Ž A . be the set of all minimal elements of BŽ A .. Then sq Ž M Ž A . . s t q 2 r F t q
2 Ž t y 1.
ay2
for some r g 0 4 j N. Ž 3.1.
Moreo¨ er, if Ž t y 1.rŽ a y 2. is a positi¨ e integer, then there exists an A 9 g LCJa Ž n, t . with < A 9 < s Na Ž n, t . such that sq Ž M Ž A 9 . . s t q 2 r X - t q
2 Ž t y 1.
ay2
for some r X g 0 4 j N. Ž 3.2.
Remark 1. From this result easily follows the proof of conjecture Ž1.6., stated in the Introduction. The proof goes as follows: For n F t q 1 or a G t q 1, we have from Ž3.2. sqŽ M Ž AX .. s t and hence < A < s Na Ž n, t . s
438
AHLSWEDE AND KHACHATRIAN
a ny t s < K 0 < and if n ) t q 1 and a - t q 1 it is easily verified that Na Ž n, t . G < K 1 < ) < K 0 < . Remark 2. From this result the following observation is immediate: The structure of the optimal families does not depend on n for n G t q 2Ž t y 1.rŽ a y 2.. Remark 3. The proof of Lemma 6 is mainly based on the ideas and methods used in our previous paper w3x. Proof. First we prove Ž3.1.. Assume to the opposite of Ž3.1. that sq Ž M Ž A . . s l ) t q
2 Ž t y 1.
ay2
Ž 3.3.
or lFtq
2 Ž t y 1.
ay2
2 ¦ l y t.
but
Ž 3.4.
We shall show that under these assumptions there exists an A 9 g Ja Ž n, t . with < A 9 < ) < A <, which is a contradiction to < A < s Na Ž n, t .. For this we start with the partition
˙ M1 Ž A . , M Ž A . s M0 Ž A . j where M0 Ž A . s E g M Ž A . : sq Ž E . s sq Ž M Ž A . . s l 4 and M1 Ž A . s M Ž A . _ M 0 Ž A . . Obviously, for every E1 g M0 Ž A . and E2 g M1Ž A ., we have
Ž E1 _ l 4 . l E2
G t.
The elements in M0 Ž A . have an important property, which follows immediately from Lemma 5: ŽP. For any E1 , E2 g M0 Ž A . with < E1 l E2 < s t necessarily, < E1 < q < E2 < s l q t.
439
DIAMETRIC THEOREM IN HAMMING SPACES
Now we partition M0 Ž A . according to the cardinalities of its members M0 Ž A . s
D Ri ,
Ri s M0 Ž A . l
i
žw x/ n i
.
Of course, some of the Ri ’s can be empty. Next we omit the element l 4 ; that is, we consider RiX s E ; w 1, l y 1 x : E j l 4 g Ri 4 . So < Ri < s < RiX < and, for EX g RiX , < EX < s i y 1. From property ŽP. we know that, for E1X g RiX , E2X g RjX with i q j / l q t necessarily, < E1X l E2X < G t.
Ž 3.5.
We shall prove that Žunder assumptions Ž3.3. or Ž3.4.. all Ri ’s are empty. Suppose that, for some i, Ri / B or, equivalently, RiX / B. We distinguish two cases: Ža. i / Ž l q t .r2 and Žb. i s Ž l q t .r2. Case Ža.. We consider the sets f 1 s M1 Ž A . j Ž M0 Ž A . _ Ž Ri j Rlqtyi . . j RiX and X f 2 s M1 Ž A . j Ž M0 Ž A . _ Ž Ri j Rlqtyi . . j Rlqtyi .
We know already Žsee property ŽP. and Ž3.5.. that f 1 , f 2 g I Ž n, t . and hence Ai s V Ž f i . g Ja Ž n, t .
for i s 1, 2.
The desired contradiction shall take the form max < Ai < ) < A < .
is1, 2
Ž 3.6.
We consider the set A _ A1. From the construction of f 1 and Ri ’s, it follows that A _ A1 s
D
DŽ E. ,
Eg Rlqtyi
where the DŽ E .’s are defined in Ž2.8.. Using Lemma 4, we have < A _ A1 < s < Rlqtyi < ? Ž a y 1 . iyt ? a nyl .
Ž 3.7.
440
AHLSWEDE AND KHACHATRIAN
Now we consider A1 _ A. Let E1 be any element of Ri ’s, so < E1 < s i y 1. We consider DX Ž E1 . s A s Ž a1 , . . . , a n . g Xan : B Ž A . l w 1, l x s E1 and Ž a lq1 , . . . , a n . g Xany l 4
Ž 3.8.
Žrecalling that B Ž A. g 2 w nx is the set-theoretical image of A.. It can easily be seen that DX Ž E1 . g A1 _ A
Ž 3.9.
and DX Ž E1 . s Ž a y 1 .
lyiq1
? a nyl .
Ž 3.10.
We also notice that, for E1 , E2 g RiX , E1 / E2 , one has DX Ž E1 . l DX Ž E2 . s B.
Ž 3.11.
< A1 _ A < G < Ri < ? Ž a y 1 . lyiq1 ? a nyl .
Ž 3.12.
Therefore
Analogously, we have < A _ A2 < s < Ri < ? Ž a y 1 . lyi ? a nyl
Ž 3.13.
< A2 _ A1 < G < Rlqtyi < ? Ž a y 1 . iytq1 ? a nyl .
Ž 3.14.
and
Actually, it is easy to show that there are equalities in Ž3.12. and Ž3.14.. However, that is not needed here. Now Ž3.7. and Ž3.12. ] Ž3.14. enable us to state the negation of Ž3.6. in the form < Ri < Ž a y 1 . lyiq1 ? a nyl F < Rlqtyi < ? Ž a y 1 . iyt ? a nyl , < Rlqtyi < Ž a y 1 . iytq1 ? a nyl F < Ri < ? Ž a y 1 . lyi ? a nyl , which is obviously false, because Ri / B and a ) 2. Case Ža., which we have just considered, shows that 2 <Ž t q l ., l s sqŽ M Ž A ... This shows that assumption Ž3.4. is false. Moreover, if 2 < l q t, then necessarily < E< s
lqt 2
for all E g M Ž A . with sq Ž E . s sq Ž M Ž A . . s l.
DIAMETRIC THEOREM IN HAMMING SPACES
441
Case Žb.. Here necessarily 2 < l q t. We consider the set RŽXtql .r2 and recall that, for E g RŽXtql .r2 , < E< s
tql 2
y1
E ; w 1, l y 1 x .
and
By the pigeon-hole principle there exist an i g w1, l y 1x and a T ; RŽXtql .r2 such that i f E for all E g T and
lyt 2 Ž l y 1.
? < RŽXtql .r2 < .
Ž 3.15.
By Lemma 5 we have < E1 l E2 < G t for all E1 , E2 g T and since by Case Ža. Ri s B for i / Ž t q l .r2, we get f X s Ž M Ž A . _ RŽ tql .r2 . j T g I Ž n, t . and hence V Ž f X . g Ja Ž n, t . . We are going to show Žunder condition Ž3.3.. that V Ž f X . ) < A <.
Ž 3.16.
Indeed, let us write
˙ D2 , A s V Ž M Ž A . . s D1 j where D 1 s V Ž M Ž A . _ RŽ tql .r2 . , D2 s V
Ž RŽ tql.r2 . _ V Ž M Ž A . _ RŽ tql .r2 . ,
and
˙ D3 , V Ž f X . s D1 j where D 3 s V Ž T . _ V Ž M Ž A . _ RŽ tql .r2 . . In this terminology, equivalent to Ž3.16. is < D3 < ) < D2 < .
Ž 3.17.
442
AHLSWEDE AND KHACHATRIAN
We know by Lemma 4 that D 2 s < RŽ tql .r2 < ? Ž a y 1 .
Ž lyt .r2
? a nyl
Ž 3.18.
and estimate < D 3 < from below. Let E g T , E ; w l y 1x, and < E < s Ž t q l .r2 y 1. We consider C Ž E . s A s Ž a1 , . . . , a n . g Xan : B Ž A . l w l y 1 x s E and Ž a l , . . . , a n . g Xany lq1 4 . Clearly, C Ž E . g D 3 and < C Ž E .< s Ž a y 1.Ž lyt .r2 ? a nylq1. We also notice that, for all E1 , E2 g T , E1 / E2 , we have C Ž E1 . l C Ž E2 . s B. Therefore < D 3 < G < T < ? Ž a y 1 . Ž lyt .r2 ? a nylq1 .
Ž 3.19.
Actually, as in the similar situation in Case Ža., we have equality in Ž3.19., but again it is not needed here. In the light of Ž3.15. and Ž3.17. ] Ž3.19. sufficient for Ž3.16. is lyt 2 Ž l y 1.
< RŽ tql .r2 < ? Ž a y 1 . Ž lyt .r2 ? a nylq1
) < RŽ tql .r2 < ? Ž a y 1 .
Ž lyt .r2
? a nyl .
According to Ž3.3. and RŽ tql .r2 / B, this is true. Therefore Ž3.16. holds in contradiction to the optimality of A. Hence the assumption Ž3.3. is false and the first part of Lemma 6 is proved. Now let Ž t y 1.rŽ a y 2. be a positive integer and let sq Ž M Ž A . . s l s t q 2
ty1
ay2
.
Ž 3.20.
We already know from Case Ža. that < E< s
lqt 2
for all E g M Ž A . with sq Ž E . s l.
We repeat the steps described in Case Žb. and observe that instead of Ž3.16., under assumption Ž3.20., a slightly weaker inequality V Ž f X . G < A < holds. Lemma 6 is proved. We need the following ‘‘comparison lemma,’’ which makes it possible to link the theorem with Theorem AK via its corollary in Section 4.
DIAMETRIC THEOREM IN HAMMING SPACES
443
wmx
Let S ; 2 2 , that is, L g S ª L ; 2 w m x. For given t g N, t F m, and bt , btq1 , . . . , bm g Rq, we consider m
h Ž S, bt , . . . , bm . s max
Ý < Li < ? bi ,
lgS ist
Ž 3.21.
where Li s L l Ž w mi x .. Suppose there is an L U g S so that, for some r g N, LiU s B
if t F i - t q r and < LiU < G < Li <
Ž ).
for all t q r F i F m and all L g S. wmx
LEMMA 7. Let L ; 2 2 , bt , btq1 , . . . , bm g Rq, and let hŽ S, bt , . . . , bm . be assumed at L U g S, which is described just abo¨ e. Then for any g t , . . . , gm g Rq such that bi gi G , i s t , . . . , m y 1, Ž 3.22. biq1 g iq1 still hŽ S, g t , . . . , gm . is assumed at L U , that is, m
h Ž S, g t , . . . , gm . s max
Ý < Li < ? g i s
LgS ist
m
Ý < LiU < ? g i . ist
Proof. Let g t , . . . , gm g Rq, such that birbiq1 G g irg iq1 for i s t, . . . , m y 1 and let us prove that m
m
ist
ist
Ý < LiU < ? g i G Ý < Li < g i
Ž 3.23.
for every L g S. From the condition on L U in the lemma, we know that m
m
ist
ist
Ý < LiU < bi G Ý < Li < ? bi .
Ž 3.24.
Without loss of generality, we can assume bm s gm s 1. We write the numbers bt , . . . , bm , g t , . . . , gm in the form:
bm s 1
gm s 1
bmy 1 s dmy1
gmy1 s « my1
bmy 2 s dmy1 ? dmy2 .. . bi s dmy1 ? dmy2 ??? d i ??? bt s dmy1 dmy2 ??? d t
gmy2 s « my1 « my2 .. . g i s « my1 « my2 ??? « i ??? g t s « my1 « my2 ??? « t .
444
AHLSWEDE AND KHACHATRIAN
Then condition Ž3.22. is equivalent to the inequality
di G « i ,
i s t , . . . , m y 1.
Ž 3.25.
Let l g N be the smallest integer for which dmy 1 s « my1 , dmy2 s « my 2 , . . . , dmylq1 s « mylq1 , but dmyl ) « myl . X Now we consider btX , . . . , bmX , where bmX s bm s 1, . . . , bmylq1 s bmylq1 X and bi s bi ? « mylrdmyl , t F i F m y l. Let us show that m
m
ist
ist
Ý < LiU < biX G Ý < Li < ? biX .
Ž 3.26.
If m y l q 1 F t q r, then the condition Ž). and Ž3.24. imply m
m
m
m
Ý < LiU < ? biX s
Ý < LiU < ? bi G
Ý < Li < ? bi G
Ý < Li < ? biX .
ist
ist
ist
ist
If m y l q 1 ) t q r, then Ž3.26. is equivalent to m
< LiU < ? bi q
Ý ismylq1
< LiU < ?
Ý istqr
m
G
myl
< Li < ? bi q
Ý ismylq1
bi « myl dmy l
myl
Ý < Li < ?
ist
bi « myl dmy l
,
which, in turn, is equivalent to m
Ž dmy l y « myl . ?
< LiU < ? bi q « myl ?
Ý ismylq1
< LiU < ? bi
istqr m
G Ž dmy l y « myl . ?
m
Ý
< Li < ? bi q « myl
Ý ismylq1
m
Ý < Li < ? bi ist
and finally to m
Ž dmy l y « myl . ?
Ý
Ž < LiU < y < Li < . bi
ismylq1
q « my l
ž
m
m
ist
ist
Ý < LiU < ? bi y Ý < Li < ? bi
/
G 0.
This is true because dmy l ) « myl , < LiU < G < Li < for i G m y l q 1 ) t q r Žsee condition Ž). and Ž3.24...
445
DIAMETRIC THEOREM IN HAMMING SPACES
It is easily seen that, continuing this transformation, we will arrive at the coefficients g t , . . . , gm and Ž3.23. holds. Remark. Lemma 7 can be formulated for much more general strucwmx m tures. For instance, instead of 2 2 one can take Rq s L s Ž l 1 , . . . , l m .: U m l i G 04 , choose suitable S ; Rq , L g S, and the claim of Lemma 7 still holds.
4. PROOF OF THE THEOREM At first let us recall the sets
w nx
½ ž /
Fr s F g
k
: F l w 1, t q 2 r x G t q r
5
and Kr s A g Xan : B Ž A . l w 1, t q 2 r x G t q r 4 , where B Ž A. g 2 w nx is the set-theoretical image of A. Let us also define the set D Ž r , t . s D g 2 w tq2 r x : < D < G t q r 4 and let Di s D l
žw
t q 2r x . i
/
We note that < Di < s 0 if i - t q r, and < Di < s Ž t qi 2 r . if i G t q r. Clearly, D Ž r, t . g I Ž t q 2 r, t .. We can write the cardinalities of Fr and Kr as follows: < Fr < s
r
Ý js0 tq2 r
s
Ý is0
ž
2r q t n y 2r y t ? tqrqj kytyryj
/ž
< Di < ? n y 2 r y t kyi
ž
/
/
and < Kr < s
r
Ý js0
ž
2r q t ryj ? Ž a y 1. ? a ny2 ryt tqrqj
/
tq2 r
s a ny2 ryt ?
Ý is0
< Di < ? Ž a y 1 . 2 rqtyi .
446
AHLSWEDE AND KHACHATRIAN
Now we present an easy but important consequence of Theorem AK stated in the Introduction. COROLLARY.
Let
Ž k y t q 1. 2 q
ž
ty1 rq1
/
- m - Ž k y t q 1. 2 q
ž
ty1 r
/
r g N,
,
Ž 4.1. and let
gi s
ž
m y 2r y t kyi
/
for i G t.
Ž 4.2.
Then 2 rqt
max
Ý
AgI Ž2 rqt , t . ist
< Ai < ? g i ,
where Ai s A l
žw
2r q tx , i
/
is assumed at A s D Ž r, t .. Now we are ready to prove our main result. Proof of the theorem. Let r g 04 j N be the biggest integer so that
½
t q 2 r - min n q 1, t q 2
ty1
ay2
5
.
Ž 4.3.
From Lemma 6 it immediately follows that, for this r, tq2 r
Na Ž n, t . s
max
Ý
AgI Ž2 rqt , t . ist
< Ai < ? Ž a y 1 . tq2 ryi ? a nyty2 r
tq2 r
sc?
max
Ý
AgI Ž2 rqt , t . ist
< Ai < ? Ž a y 1 . tq2 ryi ,
Ž 4.4.
where c s a ny ty2 r is a constant. We note that in the case n F t q 2Ž t y 1.rŽ a y 2. we have c s 1 or c s a . Now we are going to apply Lemma 7 with respect to m s t q 2 r, w2 tq r x 2r y t. S s I Ž t q 2 r, t . ; 2 2 , g i s Ž m0 y , i s t, . . . , t q 2 r, where m 0 is an ky i integer from the interval in Ž4.1., d i s Ž a y 1. tq2 ryi Žsee Ž4.4... As a set L U in Lemma 7 we take L U s D Ž r, t . g I Ž t q 2 r, t ., since D Ž r, t . has the properties ŽU . and, according to the corollary, satisfies the condition in Lemma 7 on the set L U .
447
DIAMETRIC THEOREM IN HAMMING SPACES
To apply Lemma 7, it remains to show the existence of a suitable m 0 from the interval in Ž4.1. for which the condition Ž3.22. in Lemma 7 holds:
gi g iq1
G Ž a y 1. s
di d iq1
i s t , . . . , t q 2 r y 1.
,
Ž 4.5.
For this, necessarily we must have k G t q 2r
Ž 4.6.
and m0 G a Ž k y t . q 2 r q t y 1
Ž this follows from Ž 4.5. . . Ž 4.7.
Hence, to apply Lemma 7, it remains to show the existence of an integer k g N for which the system of inequalities
Ž k y t q 1. 2 q
ž
ty1 rq1
/
- m0 - Ž k y t q 1. 2 q
ž
ty1 r
/
,
Ž 4.8.
a Ž k y t . q 2 r q t y 1 F m0 is solvable for m 0 g N provided that the conditions k G t q 2r
r-
and
ty1
ay2
Ž 4.9.
hold Žsee Ž4.6. and Ž4.3... The system Ž4.8. is equivalent to
¡
rm 0
~ 2r q t y 1
q t y 1-k -
¢
m0 Ž r q 1.
q t y 1, 2r q t q 1 m0 2r q t y 1 kF y q t. a a
Ž 4.10.
To guarantee the existence of a k g N for the first inequality, it is sufficient to take m 0 so big that rm 0 2r q t y 1
qty1-
m0 Ž r q 1. 2r q t q 1
q t y 2,
which gives m0 )
Ž 2 r q t q 1. Ž 2 r q t y 1. ty1
.
Ž 4.11.
448
AHLSWEDE AND KHACHATRIAN
Now we consider the inequality rm 0 2r q t y 1
qty1-
m0
a
y
2r q t y 1
a
qty1
or, equivalently, 2 Ž 2 r q t y 1. - m0 Ž 2 r q t y 1 y a r . .
Since r - Ž t y 1.rŽ a y 2. Žsee condition Ž4.9.. and, consequently, 2 r q t y 1 y b r ) 0, we have m0 )
Ž 2 r q t y 1.
2
2r q t y 1 y a r
.
Ž 4.12.
Next we consider the inequality 2r q t -
rm 0 2r q t y 1
q t y 1,
which gives m0 )
Ž 2 r q 1. Ž 2 r q t y 1. r
.
Ž 4.13.
Finally, we take m 0 g N so big, that m 0 satisfies Ž4.11. ] Ž4.13., and we take as k g N the smallest integer such that k ) rm 0rŽ2 r q t y 1. q t y 1. For these m 0 , k g N, Ž4.10. and Ž4.9. hold. Consequently, Ž4.8. also holds. Hence g irŽg i q 1. G Ž a y 1. s d irŽ d i q 1. for all i s t, . . . , t q 2 r y 1 and we can apply Lemma 7. This finishes the proof of the theorem because Na Ž n, t . s < Kr < s a ny2 ryt ?
tq2 r
Ý
< Di < ? Ž a y 1 . 2 rqtyi ,
is0
where Di s D Ž t, r . l Ž w t qi 2 r x ..
REFERENCES 1. R. Ahlswede, N. Cai, and Z. Zhang, Diametric theorems in sequence spaces, Combinatorica 12Ž1. Ž1992., 1]17. 2. R. Ahlswede and G. O. H. Katona, Contributions to the geometry of Hamming spaces, Discrete Math. 17 Ž1977., 1]22. 3. R. Ahlswede and L. H. Khachatrian, The complete intersection theorem for systems of finite sets, European J. Combin., to appear.
DIAMETRIC THEOREM IN HAMMING SPACES
449
4. C. Berge, Nombres de coloration de l’hypergraphe h-parti complet, in ‘‘Proceedings of the Hypergraph Seminar, Columbus, Ohio, 1972,’’ pp. 13]20, Springer-Verlag, New York, 1974. 5. P. Erdos, ˝ C. Ko, and R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. 12 Ž1961., 313]320. 6. P. Frankl, The Erdos]Ko]Rado theorem is true for n s ckt, in ‘‘Proceedings of the Fifth ˝ Hungarian Combinatorics Colloquinn Keszthely, 1976,’’ pp. 365]375, North-Holland, Amsterdam, 1978. 7. P. Frankl and Z. Furedi, The Erdos]Ko]Rado theorem for integer sequences, SIAM J. ¨ ˝ Algebraic Discrete Methods 1Ž4. Ž1980., 316]381. 8. G. O. H. Katona, Intersection theorems for systems of finite sets, Acta Math. Hungar. 15 Ž1964., 329]337. 9. D. J. Kleitman, On a combinatorial conjecture of Erdos, ˝ J. Combin. Theory 1 Ž1966., 209]214. 10. M. L. Livingston, An ordered version of the Erdos]Ko]Rado theorem, J. Combin. Theory ˝ Ser. A 26 Ž1979., 162]165. 11. A. Moon, An analogue of Erdos]Ko]Rado theorem for the Hamming schemes H Ž n, q ., ˝ J. Combin. Theory Ser. A 32 Ž1982., 386]390. 12. R. M. Wilson, The exact bound in the Erdos]Ko]Rado theorem, Combinatorica 4 Ž1984., ˝ 247]257.
20, 429]449 Ž1998.
AM980588
The Diametric Theorem in Hamming Spaces}Optimal Anticodes Rudolf Ahlswede and Levon H. Khachatrian Mathematik, Uni¨ ersitaat Bielefeld, Bielefeld 33501, Germany Received July 1, 1995; accepted April 21, 1997
For a Hamming space Ž Xan , d H ., the set of n-length words over the alphabet Xa s 0, 1, . . . , a y 14 endowed with the distance d H , which for two words x n s Ž x 1 , . . . , x n ., y n s Ž y 1 , . . . , yn . g Xan counts the number of different components, we determine the maximal cardinality of subsets with a prescribed diameter d or, in another language, anticodes with distance d. We refer to the result as the diametric theorem. In a sense anticodes are dual to codes, which have a prescribed lower bound on the pairwise distance. It is a hopeless task to determine their maximal sizes exactly. We find it remarkable that the diametric theorem Žfor arbitrary a . can be derived from our recent complete intersection theorem, which can be viewed as a diametric theorem Žfor a s 2. in the restricted case, where all n-length words considered have exactly k ones. Q 1998 Academic Press
1. PREVIOUS RESULTS, CONJECTURES, AND THE NEW THEOREM This paper is another demonstration of the power of the methods of w3x. We stick to the earlier notation as far as possible and first repeat it. Then we state the complete intersection theorem in its historical context, because this enables us to put the new result into proper perspective. Here we need some more terminology for the formulation of known results and conjectures for the diametric problem in Hamming space or related intersection problems. Finally, we state the new diametric theorem. N denotes the set of positive integers and for i, j g N, i - j, the set i, i q 1, . . . , j4 is abbreviated as w i, j x. Moreover, for w1, j x we also write w j x. 429 0196-8858r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
430
AHLSWEDE AND KHACHATRIAN
For k, n g N, k F n, we set 2 w nx s F : F ; w 1, n x 4
žw x/ n k
and
s F g 2 w nx : < F < s k 4 .
A system of sets A ; 2 w nx is called t-intersecting, if < A1 l A 2 < G t
for all A1 , A 2 g A ,
Ž 1.1.
and I Ž n, t . denotes the set of all such systems. Moreover, we define I Ž n, k, t . s A g I Ž n, t .: A ; Ž w kn x .4 . The investigation of the function M Ž n, k, t . s max A g IŽ n, k, t . < A <, 1 F t F k F n, and the structure of maximal systems was initiated by Erdos, ˝ Ko, and Rado w5x. THEOREM EKR w5x.
For 1 F t F k and n G n 0 Ž k, t . Ž suitable., M Ž n, k, t . s
ž
nyt . kyt
/
Clearly, the system
w nx
½ ž /
A Ž n, k, t . s A g
k
: w 1, t x ; A
5
y t. Ž . is t-intersecting, has cardinality Ž nk y t , and is therefore optimal for n G n 0 k, t .
The smallest n 0 Ž k, t ., for which this is the case, has been determined by Frankl w6x for t G 15 and subsequently by Wilson w12x for all t: n0 Ž k , t . s Ž k y t q 1. Ž t q 1. . In the recent paper w3x we have settled all the remaining cases: n Ž k y t q 1.Ž t q 1.. The following result plays a key role in the present paper. COMPLETE INTERSECTION THEOREM AK w3x. Define Fi s F g Ž w kn x .: < F l w1, t q 2 i x< G t q i4 for 0 F i F Ž n y t .r2. For 1 F t F k F n with
Ž i. Ž k y t q 1. 2 q
ž
ty1 rq1
/
- n - Ž k y t q 1. 2 q
ž
ty1 r
/
for some r g N j 0 4
431
DIAMETRIC THEOREM IN HAMMING SPACES
we ha¨ e M Ž n, k, t . s < Fr < and Fr is}up to permutations}the unique optimum. By con¨ ention, Ž t y 1.rr s ` for r s 0.
Ž ii .
ty1
Ž k y t q 1. 2 q
ž
rq1
/
sn
for r g N j 0 4
we ha¨ e M Ž n, k, t . s < Fr < s < Frq1 < and an optimal system equals}up to permutations}either Fr or Frq1. Erdos, ˝ Ko, and Rado also initiated the study of optimal systems in I Ž n, t . and of the function M Ž n, t . s
max < A < .
AgI Ž n , t .
A complete description was given by Katona, who, in particular, obtained THEOREM Ka w8x.
¡
M Ž n, t . s
~
n
Ý
is Ž nqt .r2
n , i
n
¢
if n q t is e¨ en,
ž /
Ý
is Ž nqtq1 .r2
n q i
ž /
ny1 nqty1 2
0
,
if n q t is odd.
His proof proceeds by estimating ‘‘shadows’’ of sets in Ž w kn x .. Actually, it also can be proved by the method of the present paper. Now we make a transition from 2 w nx to X 2n s 0, 14 n and the more general Xan s 0, 1, . . . , a y 14 n. Clearly, any set A g 2 w nx can be represented as word a n s Ž a1 , . . . , a n . g X 2n, where ai s
½
1, 0,
if i g A, if i g w n x _ A,
and conversely. Furthermore, in Xan we have a second concept of intersection. We call A ; Xan t y Xa-intersecting, if, for all a n, b n g A, int Ž a n , b n . J j g w n x : a j s bj 4 G t. Let Ja Ž n, t . denote the set of all such systems.
Ž 1.2.
432
AHLSWEDE AND KHACHATRIAN
Since d H Ž a n, b n . s Ž j g w1, n x: a j / bj 4. s n y intŽ a n, b n ., we can equivalently say that A has a diameter diam Ž A . J
max d H Ž a n , b n . F d s n y t
a n , b ng A
or that A is d-diametric. It is important to notice that the notions t y X 2-intersecting in X 2n and t-intersecting in 2 w nx are quite different! We are concerned here with the function Na Ž n, t . s
max
AgJa Ž n , t .
< A <.
Ž 1.3.
There is already a well-known result for a s 2 due to Kleitman: THEOREM K1 w9x.
¡
Ž nyt .r2
N2 Ž n, t . s
~
¢2
n , i
ž / Ý ž
Ý
is0
Ž nyty1 .r2 is0
if n y t is e¨ en, ny1 , i
/
if n y t is odd.
This result and Theorem Ka imply N2 Ž n, t . s M Ž n, t . .
Ž 1.4.
Actually, it was shown in w2x that the two theorems can be easily derived from each other by passing through upsets. Finally, we report on the results known Žto us., which were obtained during the last three decades on Na Ž n, t . for a ) 2. Berge w4x proved that Na Ž n, 1 . s a ny 1
for a G 3.
Ž 1.5.
Livingston w10x showed that the A g Ja Ž n, 1. satisfying Ž1.5. are of the form A s a n s Ž a1 , . . . , a n . g Xan : a i s a4 w7x conjecfor some i g w n x and a g 0, 1, . . . , a y 14 . Frankl and Furedi ¨ tured that Na Ž n, t . s a ny t
iff n F t q 1 or a G t q 1,
Ž 1.6.
and they proved this for t G 15. Ahlswede, et al. w1x remarked that, for
433
DIAMETRIC THEOREM IN HAMMING SPACES
n F t q 1 q log trlogŽ a y 1.,
¡
Ž nyt .r2
Na Ž n, t . s
~
¢a
Ý
n i
i
ž /Ža . Ý ž /Ža y1 ,
is0 Ž nyty1 .r2
ny1 i
is0
if n y t is even,
Ž 1.7. i
y 1. ,
if n y t is odd.
w7x Žand in a diametric formulation also Ahlswede et al. Frankl and Furedi ¨ w1x. have made the General Conjecture. Na Ž n, t . s
max
0FiF Ž nyt .r2
< Ki <
for all n, a , t ,
Ž 1.8.
where, with the convention B Ž a n . s j: a j s a y 14 , Ki s a n g Xan : B Ž a n . l w 1, t q 2 i x G t q i 4 .
Ž 1.9.
Clearly, Ki g Ja Ž n, t . for 0 F i F Ž n y t .r2. We note that in this terminology the results Ž1.6. and Ž1.7. can be summarized in the form
¡< K < , N Ž n, t . s~ ¢< K
if n F t q 1 or a G t q 1, t G 15, log t if n F t q 1 q . log Ž a y 1 .
0
a
?Ž nyt .r2 @
<,
This covers only very few values of the parameters n, a , t. We settle here all cases by establishing the general conjecture. DIAMETRIC THEOREM. such that
For a G 2 let r g 04 j N be the largest integer
½
t q 2 r - min n q 1, t q 2
ty1
ay2
5
.
Then Na Ž n, t . s < Kr <. Ž By con¨ ention, Ž t y 1.rŽ s y 2. s ` for a s 2.. Remark 1. Actually, we also can prove that, up to permutation of 1, 2, . . . , n4 and permutations of the alphabet in the components, there is exactly one optimal configuration, unless t ) 1, t q 2Ž t y 1.rŽ a y 2. F n and Ž t y 1.rŽ a y 2. is integral, in which case we have two optimal configurations, KŽ ty1.rŽ ay2. and KŽ ty1.rŽ ay2.y1.
434
AHLSWEDE AND KHACHATRIAN
Remark 2. A generalization of Theorem Ka to every a G 2 can be obtained with the intersection concept based on the quantity intU Ž a n , b n . J j g w n x : a j s bj s a y 1 4 instead of the quantity intŽ a n, b n . defined in Ž1.2.. In analogy to Na Ž n, t . we get now Ma Ž n, t ., where M2 Ž n, t . equals the familiar M Ž n, t . and obviously ?Ma Ž n, t . F Na Ž n, t .. The structure of the Ki ’s and the diametric theorem imply now Ma Ž n, t . s Na Ž n, t .. In particular, we have thus shown that Theorem Ka can also be proved by our methods.
2. REDUCTION TO CANONICAL AND STABLE SETS We give combinatorial characterizations of Na Ž n, t . of increasing precision. The first one is not new. PROPOSITION FF w7x. n
Na Ž n, t . s
max
Ý < Gi < Ž a y 1. ny i ,
Ž 2.1.
GgI Ž n , t . is0
where Gi s G l Ž w ni x .. We derive this result, because we want to start from first principles and at the same time introduce some concepts. Here it is more convenient to write A for a word a n s Ž a1 , . . . , a n .. It seems that the following transformation was first used by Kleitman w9x. For any A ; Xan, any A s Ž a1 , a2 , . . . , a n . g A, and 1 F j F n, 0 F i F a y 1, we define
¡Ž a , . . . , a , a y 1, a , . . . , a . , this is not element of A and a s i , ¢A, ifotherwise
T Ž A . s~ ji
1
jy1
jq1
n
j
and Tji Ž A . s Tji Ž A . : A g A 4 . Repeated application of these transformations yields after finitely many steps an AX ; Xan, for which Tji Ž AX . s AX
for all 1 F j F n, 0 F i F a y 1.
Ž 2.2.
435
DIAMETRIC THEOREM IN HAMMING SPACES
DEFINITION 2.1. A set A ; Xan is said to be canonical, if for all 1 F j F n, 0 F i F a y 1.
Ti j Ž A . s A
The transformation Tji has the important properties to keep the cardinality and the t y Xa-intersection property unchanged; that is, < Tji Ž A .< s < A < and A g Ja Ž n, t . implies Tji Ž A . g Ja Ž n, t .. Hence Na Ž n, t . s
max
AgJa Ž n , t .
< A< s
max
AgCJa Ž n , t .
< A <,
Ž 2.3.
where CJa Ž n, t . ; Ja Ž n, t . is the set of all canonical systems in Ja Ž n, t .. Now we make the transition to 2 w nx. DEFINITION 2.2. To a system A g CJa Ž n, t . we associate the set-theoretical image BŽ A . s B Ž A.: A g A 4 , where B Ž A. is defined in Ž1.8.. We have an immediate consequence w7x.. LEMMA 1 ŽFrankl and Furedi ¨
For A g CJa Ž n, t .,
B Ž A . g I Ž n, t . . DEFINITION 2.3. For any D g 2 w nx, we define the upset U Ž D . s DX g 2 : D ; DX 4 . More generally Žwith slight abuse of notation. for C ; 2 w nx, we define the upset w nx
U Ž D. s
D U Ž D. .
Dg D
Again, we have a direct consequence of the definitions. LEMMA 2. Let A g CJa Ž n, t . satisfy < A < s Na Ž n, t ., and let BŽ A . be the set-theoretical image of A. Then Ži.
Ž ii .
BŽ A . is an upset. < A< s
Ý
< Ž A .<
Ž a y 1 . ny B
B Ž A .g BŽ A . n
s
Ý g i Ž a y 1. ny i ,
Ž 2.4.
is0
where g i s BŽ A . l
žw x/ n i
.
This yields the proposition. We introduce another familiar concept.
436
AHLSWEDE AND KHACHATRIAN
DEFINITION 2.4. A set of subsets D ; 2 w nx is said to be left-compressed or stable, if, for every D g D and every 1 F i - j F n with i f D, j g D necessarily, DX s ŽŽ D _ j4. j i4. g D. We denote by LI Ž n, t . ; I Ž n, t . the set of all stable systems in I Ž n, t . and by LCJa Ž n, t . ; CJa Ž n, t . the set of all systems A g CJa Ž n, t . with BŽ A . g LI Ž n, t .. From Ž2.2. and the left-pushing technique of w5x, it readily follows that Na Ž n, t . s
max
AgJa Ž n , t .
< A< s
max
AgCJa Ž n , t .
< A< s
max
AgLCJa Ž n , t .
< A <.
Ž 2.5.
Next, for an E g 2 w nx we introduce V Ž E . s A g Xan : B Ž A . g U Ž E . 4 .
Ž 2.6.
V Ž E . s a ny < E < .
Ž 2.7.
Clearly,
More generally, for E ; 2 w nx we introduce V Ž E. s
D V Ž E. .
Eg E
DEFINITION 2.5. Let A g CJa Ž n, t . and let BŽ A . be the set-theoretical image of A. Then A is called a-upset, if AsV
Ž BŽ A . . .
DEFINITION 2.6. For E s e1 , e2 , . . . , e < E < 4 ; w n x, e1 - e2 - ??? - e < E < , write the biggest element e < E < as sqŽ E .. Also for E ; 2 w nx set sq Ž E . s max sq Ž E . . Eg E
The next important properties immediately follow from left-compressedness arguments Žsimilar to those in w3x.. LEMMA 3. Let A g LCJa Ž n, t ., let A be an a-upset, and let BŽ A . be the set-theoretical image of A. Further, let M Ž A . be the set of minimal elements of BŽ A . Ž in the sense of set-theoretical inclusion.. Then A is a disjoint union As
D
EgM Ž A .
DŽ E. ,
437
DIAMETRIC THEOREM IN HAMMING SPACES
where D Ž E . s A s Ž a1 , . . . , a n . g Xan : B Ž A . l 1, sq Ž E . s E and Ž a sq Ž E .q1 , . . . , a n . g Xany s
q
Ž E.
4.
Ž 2.8.
LEMMA 4. For an a-upset A g LCJa Ž n, t ., choose E g M Ž A . such that sqŽ E . s sqŽ M Ž A .. and consider the set of elements of A, which are only generated by E, that is, AE s V Ž E . _ V Ž M Ž A . _ E . . Then AE s D Ž E .
Ž D Ž E . is defined in Ž 2.8. .
and < AE < s Ž a y 1 . s
qŽ
E .y < E <
? a nys
q
Ž E.
.
Ž 2.9.
LEMMA 5. Let A g LCJa Ž n, t ., let A be an a-upset, and let E1 , E2 g M Ž A . ha¨ e the properties i f E1 j E2 , j g E1 l E2 for some i, j g w n x with i - j. Then < E1 l E2 < G t q 1.
3. THE TWO MAIN AUXILIARY RESULTS LEMMA 6. For a ) 2 let A g LCJa Ž n, t . with < A < s Na Ž n, t ., let BŽ A . be the set-theoretical image of A, and let M Ž A . be the set of all minimal elements of BŽ A .. Then sq Ž M Ž A . . s t q 2 r F t q
2 Ž t y 1.
ay2
for some r g 0 4 j N. Ž 3.1.
Moreo¨ er, if Ž t y 1.rŽ a y 2. is a positi¨ e integer, then there exists an A 9 g LCJa Ž n, t . with < A 9 < s Na Ž n, t . such that sq Ž M Ž A 9 . . s t q 2 r X - t q
2 Ž t y 1.
ay2
for some r X g 0 4 j N. Ž 3.2.
Remark 1. From this result easily follows the proof of conjecture Ž1.6., stated in the Introduction. The proof goes as follows: For n F t q 1 or a G t q 1, we have from Ž3.2. sqŽ M Ž AX .. s t and hence < A < s Na Ž n, t . s
438
AHLSWEDE AND KHACHATRIAN
a ny t s < K 0 < and if n ) t q 1 and a - t q 1 it is easily verified that Na Ž n, t . G < K 1 < ) < K 0 < . Remark 2. From this result the following observation is immediate: The structure of the optimal families does not depend on n for n G t q 2Ž t y 1.rŽ a y 2.. Remark 3. The proof of Lemma 6 is mainly based on the ideas and methods used in our previous paper w3x. Proof. First we prove Ž3.1.. Assume to the opposite of Ž3.1. that sq Ž M Ž A . . s l ) t q
2 Ž t y 1.
ay2
Ž 3.3.
or lFtq
2 Ž t y 1.
ay2
2 ¦ l y t.
but
Ž 3.4.
We shall show that under these assumptions there exists an A 9 g Ja Ž n, t . with < A 9 < ) < A <, which is a contradiction to < A < s Na Ž n, t .. For this we start with the partition
˙ M1 Ž A . , M Ž A . s M0 Ž A . j where M0 Ž A . s E g M Ž A . : sq Ž E . s sq Ž M Ž A . . s l 4 and M1 Ž A . s M Ž A . _ M 0 Ž A . . Obviously, for every E1 g M0 Ž A . and E2 g M1Ž A ., we have
Ž E1 _ l 4 . l E2
G t.
The elements in M0 Ž A . have an important property, which follows immediately from Lemma 5: ŽP. For any E1 , E2 g M0 Ž A . with < E1 l E2 < s t necessarily, < E1 < q < E2 < s l q t.
439
DIAMETRIC THEOREM IN HAMMING SPACES
Now we partition M0 Ž A . according to the cardinalities of its members M0 Ž A . s
D Ri ,
Ri s M0 Ž A . l
i
žw x/ n i
.
Of course, some of the Ri ’s can be empty. Next we omit the element l 4 ; that is, we consider RiX s E ; w 1, l y 1 x : E j l 4 g Ri 4 . So < Ri < s < RiX < and, for EX g RiX , < EX < s i y 1. From property ŽP. we know that, for E1X g RiX , E2X g RjX with i q j / l q t necessarily, < E1X l E2X < G t.
Ž 3.5.
We shall prove that Žunder assumptions Ž3.3. or Ž3.4.. all Ri ’s are empty. Suppose that, for some i, Ri / B or, equivalently, RiX / B. We distinguish two cases: Ža. i / Ž l q t .r2 and Žb. i s Ž l q t .r2. Case Ža.. We consider the sets f 1 s M1 Ž A . j Ž M0 Ž A . _ Ž Ri j Rlqtyi . . j RiX and X f 2 s M1 Ž A . j Ž M0 Ž A . _ Ž Ri j Rlqtyi . . j Rlqtyi .
We know already Žsee property ŽP. and Ž3.5.. that f 1 , f 2 g I Ž n, t . and hence Ai s V Ž f i . g Ja Ž n, t .
for i s 1, 2.
The desired contradiction shall take the form max < Ai < ) < A < .
is1, 2
Ž 3.6.
We consider the set A _ A1. From the construction of f 1 and Ri ’s, it follows that A _ A1 s
D
DŽ E. ,
Eg Rlqtyi
where the DŽ E .’s are defined in Ž2.8.. Using Lemma 4, we have < A _ A1 < s < Rlqtyi < ? Ž a y 1 . iyt ? a nyl .
Ž 3.7.
440
AHLSWEDE AND KHACHATRIAN
Now we consider A1 _ A. Let E1 be any element of Ri ’s, so < E1 < s i y 1. We consider DX Ž E1 . s A s Ž a1 , . . . , a n . g Xan : B Ž A . l w 1, l x s E1 and Ž a lq1 , . . . , a n . g Xany l 4
Ž 3.8.
Žrecalling that B Ž A. g 2 w nx is the set-theoretical image of A.. It can easily be seen that DX Ž E1 . g A1 _ A
Ž 3.9.
and DX Ž E1 . s Ž a y 1 .
lyiq1
? a nyl .
Ž 3.10.
We also notice that, for E1 , E2 g RiX , E1 / E2 , one has DX Ž E1 . l DX Ž E2 . s B.
Ž 3.11.
< A1 _ A < G < Ri < ? Ž a y 1 . lyiq1 ? a nyl .
Ž 3.12.
Therefore
Analogously, we have < A _ A2 < s < Ri < ? Ž a y 1 . lyi ? a nyl
Ž 3.13.
< A2 _ A1 < G < Rlqtyi < ? Ž a y 1 . iytq1 ? a nyl .
Ž 3.14.
and
Actually, it is easy to show that there are equalities in Ž3.12. and Ž3.14.. However, that is not needed here. Now Ž3.7. and Ž3.12. ] Ž3.14. enable us to state the negation of Ž3.6. in the form < Ri < Ž a y 1 . lyiq1 ? a nyl F < Rlqtyi < ? Ž a y 1 . iyt ? a nyl , < Rlqtyi < Ž a y 1 . iytq1 ? a nyl F < Ri < ? Ž a y 1 . lyi ? a nyl , which is obviously false, because Ri / B and a ) 2. Case Ža., which we have just considered, shows that 2 <Ž t q l ., l s sqŽ M Ž A ... This shows that assumption Ž3.4. is false. Moreover, if 2 < l q t, then necessarily < E< s
lqt 2
for all E g M Ž A . with sq Ž E . s sq Ž M Ž A . . s l.
DIAMETRIC THEOREM IN HAMMING SPACES
441
Case Žb.. Here necessarily 2 < l q t. We consider the set RŽXtql .r2 and recall that, for E g RŽXtql .r2 , < E< s
tql 2
y1
E ; w 1, l y 1 x .
and
By the pigeon-hole principle there exist an i g w1, l y 1x and a T ; RŽXtql .r2 such that i f E for all E g T and
lyt 2 Ž l y 1.
? < RŽXtql .r2 < .
Ž 3.15.
By Lemma 5 we have < E1 l E2 < G t for all E1 , E2 g T and since by Case Ža. Ri s B for i / Ž t q l .r2, we get f X s Ž M Ž A . _ RŽ tql .r2 . j T g I Ž n, t . and hence V Ž f X . g Ja Ž n, t . . We are going to show Žunder condition Ž3.3.. that V Ž f X . ) < A <.
Ž 3.16.
Indeed, let us write
˙ D2 , A s V Ž M Ž A . . s D1 j where D 1 s V Ž M Ž A . _ RŽ tql .r2 . , D2 s V
Ž RŽ tql.r2 . _ V Ž M Ž A . _ RŽ tql .r2 . ,
and
˙ D3 , V Ž f X . s D1 j where D 3 s V Ž T . _ V Ž M Ž A . _ RŽ tql .r2 . . In this terminology, equivalent to Ž3.16. is < D3 < ) < D2 < .
Ž 3.17.
442
AHLSWEDE AND KHACHATRIAN
We know by Lemma 4 that D 2 s < RŽ tql .r2 < ? Ž a y 1 .
Ž lyt .r2
? a nyl
Ž 3.18.
and estimate < D 3 < from below. Let E g T , E ; w l y 1x, and < E < s Ž t q l .r2 y 1. We consider C Ž E . s A s Ž a1 , . . . , a n . g Xan : B Ž A . l w l y 1 x s E and Ž a l , . . . , a n . g Xany lq1 4 . Clearly, C Ž E . g D 3 and < C Ž E .< s Ž a y 1.Ž lyt .r2 ? a nylq1. We also notice that, for all E1 , E2 g T , E1 / E2 , we have C Ž E1 . l C Ž E2 . s B. Therefore < D 3 < G < T < ? Ž a y 1 . Ž lyt .r2 ? a nylq1 .
Ž 3.19.
Actually, as in the similar situation in Case Ža., we have equality in Ž3.19., but again it is not needed here. In the light of Ž3.15. and Ž3.17. ] Ž3.19. sufficient for Ž3.16. is lyt 2 Ž l y 1.
< RŽ tql .r2 < ? Ž a y 1 . Ž lyt .r2 ? a nylq1
) < RŽ tql .r2 < ? Ž a y 1 .
Ž lyt .r2
? a nyl .
According to Ž3.3. and RŽ tql .r2 / B, this is true. Therefore Ž3.16. holds in contradiction to the optimality of A. Hence the assumption Ž3.3. is false and the first part of Lemma 6 is proved. Now let Ž t y 1.rŽ a y 2. be a positive integer and let sq Ž M Ž A . . s l s t q 2
ty1
ay2
.
Ž 3.20.
We already know from Case Ža. that < E< s
lqt 2
for all E g M Ž A . with sq Ž E . s l.
We repeat the steps described in Case Žb. and observe that instead of Ž3.16., under assumption Ž3.20., a slightly weaker inequality V Ž f X . G < A < holds. Lemma 6 is proved. We need the following ‘‘comparison lemma,’’ which makes it possible to link the theorem with Theorem AK via its corollary in Section 4.
DIAMETRIC THEOREM IN HAMMING SPACES
443
wmx
Let S ; 2 2 , that is, L g S ª L ; 2 w m x. For given t g N, t F m, and bt , btq1 , . . . , bm g Rq, we consider m
h Ž S, bt , . . . , bm . s max
Ý < Li < ? bi ,
lgS ist
Ž 3.21.
where Li s L l Ž w mi x .. Suppose there is an L U g S so that, for some r g N, LiU s B
if t F i - t q r and < LiU < G < Li <
Ž ).
for all t q r F i F m and all L g S. wmx
LEMMA 7. Let L ; 2 2 , bt , btq1 , . . . , bm g Rq, and let hŽ S, bt , . . . , bm . be assumed at L U g S, which is described just abo¨ e. Then for any g t , . . . , gm g Rq such that bi gi G , i s t , . . . , m y 1, Ž 3.22. biq1 g iq1 still hŽ S, g t , . . . , gm . is assumed at L U , that is, m
h Ž S, g t , . . . , gm . s max
Ý < Li < ? g i s
LgS ist
m
Ý < LiU < ? g i . ist
Proof. Let g t , . . . , gm g Rq, such that birbiq1 G g irg iq1 for i s t, . . . , m y 1 and let us prove that m
m
ist
ist
Ý < LiU < ? g i G Ý < Li < g i
Ž 3.23.
for every L g S. From the condition on L U in the lemma, we know that m
m
ist
ist
Ý < LiU < bi G Ý < Li < ? bi .
Ž 3.24.
Without loss of generality, we can assume bm s gm s 1. We write the numbers bt , . . . , bm , g t , . . . , gm in the form:
bm s 1
gm s 1
bmy 1 s dmy1
gmy1 s « my1
bmy 2 s dmy1 ? dmy2 .. . bi s dmy1 ? dmy2 ??? d i ??? bt s dmy1 dmy2 ??? d t
gmy2 s « my1 « my2 .. . g i s « my1 « my2 ??? « i ??? g t s « my1 « my2 ??? « t .
444
AHLSWEDE AND KHACHATRIAN
Then condition Ž3.22. is equivalent to the inequality
di G « i ,
i s t , . . . , m y 1.
Ž 3.25.
Let l g N be the smallest integer for which dmy 1 s « my1 , dmy2 s « my 2 , . . . , dmylq1 s « mylq1 , but dmyl ) « myl . X Now we consider btX , . . . , bmX , where bmX s bm s 1, . . . , bmylq1 s bmylq1 X and bi s bi ? « mylrdmyl , t F i F m y l. Let us show that m
m
ist
ist
Ý < LiU < biX G Ý < Li < ? biX .
Ž 3.26.
If m y l q 1 F t q r, then the condition Ž). and Ž3.24. imply m
m
m
m
Ý < LiU < ? biX s
Ý < LiU < ? bi G
Ý < Li < ? bi G
Ý < Li < ? biX .
ist
ist
ist
ist
If m y l q 1 ) t q r, then Ž3.26. is equivalent to m
< LiU < ? bi q
Ý ismylq1
< LiU < ?
Ý istqr
m
G
myl
< Li < ? bi q
Ý ismylq1
bi « myl dmy l
myl
Ý < Li < ?
ist
bi « myl dmy l
,
which, in turn, is equivalent to m
Ž dmy l y « myl . ?
< LiU < ? bi q « myl ?
Ý ismylq1
< LiU < ? bi
istqr m
G Ž dmy l y « myl . ?
m
Ý
< Li < ? bi q « myl
Ý ismylq1
m
Ý < Li < ? bi ist
and finally to m
Ž dmy l y « myl . ?
Ý
Ž < LiU < y < Li < . bi
ismylq1
q « my l
ž
m
m
ist
ist
Ý < LiU < ? bi y Ý < Li < ? bi
/
G 0.
This is true because dmy l ) « myl , < LiU < G < Li < for i G m y l q 1 ) t q r Žsee condition Ž). and Ž3.24...
445
DIAMETRIC THEOREM IN HAMMING SPACES
It is easily seen that, continuing this transformation, we will arrive at the coefficients g t , . . . , gm and Ž3.23. holds. Remark. Lemma 7 can be formulated for much more general strucwmx m tures. For instance, instead of 2 2 one can take Rq s L s Ž l 1 , . . . , l m .: U m l i G 04 , choose suitable S ; Rq , L g S, and the claim of Lemma 7 still holds.
4. PROOF OF THE THEOREM At first let us recall the sets
w nx
½ ž /
Fr s F g
k
: F l w 1, t q 2 r x G t q r
5
and Kr s A g Xan : B Ž A . l w 1, t q 2 r x G t q r 4 , where B Ž A. g 2 w nx is the set-theoretical image of A. Let us also define the set D Ž r , t . s D g 2 w tq2 r x : < D < G t q r 4 and let Di s D l
žw
t q 2r x . i
/
We note that < Di < s 0 if i - t q r, and < Di < s Ž t qi 2 r . if i G t q r. Clearly, D Ž r, t . g I Ž t q 2 r, t .. We can write the cardinalities of Fr and Kr as follows: < Fr < s
r
Ý js0 tq2 r
s
Ý is0
ž
2r q t n y 2r y t ? tqrqj kytyryj
/ž
< Di < ? n y 2 r y t kyi
ž
/
/
and < Kr < s
r
Ý js0
ž
2r q t ryj ? Ž a y 1. ? a ny2 ryt tqrqj
/
tq2 r
s a ny2 ryt ?
Ý is0
< Di < ? Ž a y 1 . 2 rqtyi .
446
AHLSWEDE AND KHACHATRIAN
Now we present an easy but important consequence of Theorem AK stated in the Introduction. COROLLARY.
Let
Ž k y t q 1. 2 q
ž
ty1 rq1
/
- m - Ž k y t q 1. 2 q
ž
ty1 r
/
r g N,
,
Ž 4.1. and let
gi s
ž
m y 2r y t kyi
/
for i G t.
Ž 4.2.
Then 2 rqt
max
Ý
AgI Ž2 rqt , t . ist
< Ai < ? g i ,
where Ai s A l
žw
2r q tx , i
/
is assumed at A s D Ž r, t .. Now we are ready to prove our main result. Proof of the theorem. Let r g 04 j N be the biggest integer so that
½
t q 2 r - min n q 1, t q 2
ty1
ay2
5
.
Ž 4.3.
From Lemma 6 it immediately follows that, for this r, tq2 r
Na Ž n, t . s
max
Ý
AgI Ž2 rqt , t . ist
< Ai < ? Ž a y 1 . tq2 ryi ? a nyty2 r
tq2 r
sc?
max
Ý
AgI Ž2 rqt , t . ist
< Ai < ? Ž a y 1 . tq2 ryi ,
Ž 4.4.
where c s a ny ty2 r is a constant. We note that in the case n F t q 2Ž t y 1.rŽ a y 2. we have c s 1 or c s a . Now we are going to apply Lemma 7 with respect to m s t q 2 r, w2 tq r x 2r y t. S s I Ž t q 2 r, t . ; 2 2 , g i s Ž m0 y , i s t, . . . , t q 2 r, where m 0 is an ky i integer from the interval in Ž4.1., d i s Ž a y 1. tq2 ryi Žsee Ž4.4... As a set L U in Lemma 7 we take L U s D Ž r, t . g I Ž t q 2 r, t ., since D Ž r, t . has the properties ŽU . and, according to the corollary, satisfies the condition in Lemma 7 on the set L U .
447
DIAMETRIC THEOREM IN HAMMING SPACES
To apply Lemma 7, it remains to show the existence of a suitable m 0 from the interval in Ž4.1. for which the condition Ž3.22. in Lemma 7 holds:
gi g iq1
G Ž a y 1. s
di d iq1
i s t , . . . , t q 2 r y 1.
,
Ž 4.5.
For this, necessarily we must have k G t q 2r
Ž 4.6.
and m0 G a Ž k y t . q 2 r q t y 1
Ž this follows from Ž 4.5. . . Ž 4.7.
Hence, to apply Lemma 7, it remains to show the existence of an integer k g N for which the system of inequalities
Ž k y t q 1. 2 q
ž
ty1 rq1
/
- m0 - Ž k y t q 1. 2 q
ž
ty1 r
/
,
Ž 4.8.
a Ž k y t . q 2 r q t y 1 F m0 is solvable for m 0 g N provided that the conditions k G t q 2r
r-
and
ty1
ay2
Ž 4.9.
hold Žsee Ž4.6. and Ž4.3... The system Ž4.8. is equivalent to
¡
rm 0
~ 2r q t y 1
q t y 1-k -
¢
m0 Ž r q 1.
q t y 1, 2r q t q 1 m0 2r q t y 1 kF y q t. a a
Ž 4.10.
To guarantee the existence of a k g N for the first inequality, it is sufficient to take m 0 so big that rm 0 2r q t y 1
qty1-
m0 Ž r q 1. 2r q t q 1
q t y 2,
which gives m0 )
Ž 2 r q t q 1. Ž 2 r q t y 1. ty1
.
Ž 4.11.
448
AHLSWEDE AND KHACHATRIAN
Now we consider the inequality rm 0 2r q t y 1
qty1-
m0
a
y
2r q t y 1
a
qty1
or, equivalently, 2 Ž 2 r q t y 1. - m0 Ž 2 r q t y 1 y a r . .
Since r - Ž t y 1.rŽ a y 2. Žsee condition Ž4.9.. and, consequently, 2 r q t y 1 y b r ) 0, we have m0 )
Ž 2 r q t y 1.
2
2r q t y 1 y a r
.
Ž 4.12.
Next we consider the inequality 2r q t -
rm 0 2r q t y 1
q t y 1,
which gives m0 )
Ž 2 r q 1. Ž 2 r q t y 1. r
.
Ž 4.13.
Finally, we take m 0 g N so big, that m 0 satisfies Ž4.11. ] Ž4.13., and we take as k g N the smallest integer such that k ) rm 0rŽ2 r q t y 1. q t y 1. For these m 0 , k g N, Ž4.10. and Ž4.9. hold. Consequently, Ž4.8. also holds. Hence g irŽg i q 1. G Ž a y 1. s d irŽ d i q 1. for all i s t, . . . , t q 2 r y 1 and we can apply Lemma 7. This finishes the proof of the theorem because Na Ž n, t . s < Kr < s a ny2 ryt ?
tq2 r
Ý
< Di < ? Ž a y 1 . 2 rqtyi ,
is0
where Di s D Ž t, r . l Ž w t qi 2 r x ..
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