Binary (k,k)-designs

We introduce and investigate binary $(k,k)$-designs -- combinatorial structures which are related to binary orthogonal arrays. We derive general linear programming bound and propose as a consequence a universal bound on the minimum possible cardinality of $(k,k)$-designs for fixed $k$ and length $n$. Designs which attain our bound are investigated.


Introduction
Let F = {0, 1} be the alphabet of two symbols and F n 2 the set of all binary vectors x = (x 1 , x 2 , . . . , x n ) over F . The Hamming distance d(x, y) between points x = (x 1 , x 2 , . . . , x n ) and y = (y 1 , y 2 , . . . , y n ) from F n 2 is equal to the number of coordinates in which they differ.
In considerations of F n 2 as a polynomial metric space (cf. [8,15,17]) it is convenient to use the "inner product" (1) x, y := 1 − 2d(x, y) n instead of the distance d(x, y). The geometry in F n 2 is then related to the properties of the Krawtchouk polynomials {Q   i ( x, y ), i = 1, 2, . . . , n, are called moments of C.
The case of T consisting of even integers was introduced and considered by Bannai et al. in [2, Section 6.2] but (to the best of our knowledge) the special case of the next definition is not claimed yet.
. . , 2k}, where k ≤ n/2 is a positive integer, then C is called a (k, k)-design. In other words, C is a (k, k)-design if and only if M i = 0 for all i = 2, 4, . . . , 2k.
Thus, in this paper we focus on the special case when T consists of several consecutive even integers begiining with 2. It is clear from the definition that any (k, k)-design is also an (ℓ, ℓ)-design for every ℓ = 1, 2, . . . , k − 1.
After recalling general linear programming techniques, we will derive and investigate an universal (in sense of Levenshtein [17]) bound. More precisely, we obtain a lower bound on the quantity M(n, k) := min{|C| : C ⊂ F n 2 is a (k, k)-design}, the minimum possible cardinality of a (k, k)-design in F n 2 , as follows: The paper is organized as follows. In Section 2 we explain the relation between (k, k)designs and antipodal (2k + 1)-designs. Section 3 reviews the general linear programming bound and recalls the definition of so-called adjacent (to Krawtchouk) polynomials which will be important ingredients in our approach. Section 4 is devoted to our new bound. In Section 5 we discuss (k, k)-designs which attain this bound.
2. Relations to antipodal (2k + 1)-designs Classical binary m-designs have nice combinatorial properties. It follows from Definition 2.1 that the cardinality of any m-design is divisible by 2 m . This property implies a strong divisibility condition for a basic type of (k, k)-designs.
is called antipodal if for every x ∈ C the unique point y ∈ F n 2 such that d(x, y) = n (equivalently, x, y = −1) also belongs to C. The point y is denoted also by −x.
If C ⊂ F n 2 , then the (multi)set of the points, which are antipodal to points of C is denoted as usually by −C. A strong relation between antipodal (2k + 1)-designs and (k, k)-designs is given as follows.
Let the code C ⊂ F n 2 be formed by the following rule: from each pair (x, −x) of antipodal points of D exactly one of the points x and −x belongs to C. Then C is a (k, k)-design. Conversely, if C ⊂ F n 2 is a (k, k)-design which does not possess a pair of antipodal points, then D = C ∪ −C is an antipodal (2k + 1)-design in F n 2 .
Proof. For the first statement we use in (2) the antipodality of D, the relation |C| = |D|/2, and the fact that the polynomials Q for every i = 1, 2, . . . , k. Therefore C is a (k, k)-design (whichever is the way of choosing one of the points in pairs of antipodal points). The second statement follows similarly.
We note that Definition 1.2 shows that any (2k)-design is also a (k, k)-design. For small k, this relation gives some examples of (k, k)-designs with relatively small cardinalities (see Section 5).
The m-designs in F n 2 possess further nice combinatorial properties. For example, if a column of the matrix in Definition 2.1 is deleted, the resulting matrix is still an m-design in F n−1 2 with the same cardinality (possibly with repeating rows). Moreover, the rows with 0 in that column determine an (m − 1)-design in F n−1 2 of twice less cardinality. We do not know analogs of these properties for (k, k)-designs.

General linear programming bounds
Linear programming methods were introduced in coding theory by Delsarte (see [6,7]). The case of T -designs in F n 2 was recently considered by Bannai et al [2]. The transformation (1) means that all numbers x, y are rational and belong to the set T n := {−1 + 2i/n : i = 0, 1, . . . , n}. We will be interested in values of polynomials in T n .
For any real polynomial f (t) we consider its expansion in terms of Krawtchouk polynomials , where t i = −1 + 2i/n ∈ T n , i = 0, 1, . . . , n). We define the following set of polynomials The next theorem was proved (in slightly different setting) in [2]. We provide a proof here in order to make the paper self-contained.
If a (k, k)-design C ⊂ F n 2 attains this bound, then all inner products x, y of distinct x, y ∈ C are among the zeros of f (t) and f i M i (C) = 0 for every positive integer i.
Proof. Bounds of this kind follow easily from the identity (see, for example, [15,Equation (1.20)], [16,Equation (26)]), which is true for every code C ⊂ F n 2 and every polynomial f (t) = m j=0 f j Q (n) j (t). Let C be a (k, k)-design and f ∈ F n,k . We apply (4) for C and f . Since M 2j (C) = 0 for j = 1, 2, . . . , k, M i ≥ 0 for all i, and f j ≤ 0 for all odd j and for all even j > 2k, the right hand side of (4) does not exceed f 0 |C| 2 . The sum in the left hand side is nonnegative because f (t) ≥ 0 for every t ∈ T n . Thus the left hand side is at least f (1)|C| and we conclude that |C| ≥ f (1)/f 0 . Since this inequality follows for every C, we have If the equality is attained by some (k, k)-design C ⊂ F n 2 and a polynomial f ∈ F n,k , then Since f (t) ≥ 0 for every t ∈ T n , we conclude that f ( x, y ) = 0 whenever x, y ∈ C are distinct. Finally, M i (C) ≥ 0 for every i and f i ≤ 0 for i ∈ {2, 4, . . . 2k} yield f i M i (C) = 0 for every positive integer i.
We will propose suitable polynomials f (t) ∈ F n,k in the next section. Key ingredients are certain polynomials {Q 1,1 i (t)} n−2 i=0 (adjacent to the Krawtchouk ones) which were first introduced as such and investigated by Levenshtein (cf. [17] and references therein). In what follows in this section we describe the derivation of these polynomials.
The definition of the adjacent polynomials {Q 1,1 i (t)} n−2 i=0 requires a few steps as follows (cf. [17]). Let The first few (1, 1)-adjacent polynomials are Equivalently, the polynomials {Q 1,1 i (t)} n−2 i=0 can be defined as the unique series of normalized (to have value 1 at 1) polynomials orthogonal on T n with respect to the discrete measure where δ t i is the Dirac-delta measure at t i ∈ T n [17, Section 6.2]). Finally, we note the explicit formula (cf. [17, Section 6.2], [10, p. 281]) where z = n(1−t)/2, which relates the (1, 1)-adjacent polynomials and the usual (binary) Krawtchouk polynomials It follows from (9) that the polynomials Q 1,1 i (t) are odd/even functions for odd/even i (this also follows from the fact that the measure (8) is symmetric in [−1, 1]). We will use this fact when we deal with our proposal for a polynomial in Theorem 3.1.

A universal lower bound for M(n, k)
Using suitable polynomials in Theorem 3.1 we obtain the following universal bound.
If a (k, k)-design C ⊂ F n 2 attains this bound, then all inner products x, y of distinct x, y ∈ C are among the zeros of Q 1,1 Proof. We use Theorem 3.1 with the polynomial f (t) = Q 1,1 k (t) 2 of degree 2k (so we have f i = 0 for i ≥ 2k + 1) and arbitrary (k, k)-design in F n 2 . It is obvious that f (t) ≥ 0 for every t ∈ [−1, 1]. Since Q 1,1 k (t) is an odd or even function, its square is an even function. Then f i = 0 for every odd i and thus f ∈ F n,k . The calculation of the ratio f (1)/f 0 gives the desired bound.
If a (k, k)-design C ⊂ F n 2 attains the bound, then equality in (4) follows (for C and the above f (t)). Since f i M i (C) = 0 for every i, the equality |C| = f (1)/f 0 is equivalent to whence Q 1,1 k ( x, y ) = 0 whenever x and y are distinct points from C. The divisibility condition follows from Corollary 2.4. give the Rao [19] bound (see also [12,17] and references therein) for the minimum possible cardinality of (2k + 1)-designs in F n 2 , that is 2 k i=0 n−1 i . Thus our calculation of f (1)/f 0 quite resembles (and in fact follows from) the classical one [7] (see also [17,Section 2]) by noting that, obviously, the value in one is two times less and the coefficient f 0 is the same because of the symmetric measure (equivalently, since (t + 1) is equal to the sum of the odd function t Q 1,1 k (t) 2 and our polynomial).
Theorem 5.1. For fixed n and k, tight (k, k)-designs exist if and only if tight (2k + 1)designs exist.
Proof. If C ⊂ F n 2 is a tight (k, k)-design, it can not possess a pair of antipodal points since −1 is not a zero of Q 1,1 k (t). Thus we may construct an antipodal (2k − 1)-design D ⊂ F n 2 with cardinality i.e., attaining Rao bound.
Conversely, any tight (2k + 1)-design in F n 2 has cardinality 2 k i=0 n−1 i and is antipodal. By Theorem 2.3 it produces a tight (k, k)-design.
We proceed with consideration of the tight (k, k)-designs with k ≤ 3. The tight (1, 1)designs coexist with the Hadamard matrices due to a well known construction. Proof. Let C ⊂ F n 2 be a (k, k)-design with 1 + n − 1 1 = n points. Then n is divisible by 4 and, moreover, since Q 1,1 1 (t) = t, the only possible inner product is 0, meaning that the only possible distance is n/2. Therefore C is a (n, n, n/2) binary code. Changing 0 → −1 we obtain a Hadamard matrix of order n. Clearly, this works in the other direction as well.
Doubling a tight (1, 1)-design gives a tight 3-design which is clearly related to a Hadamard code (n, 2n, n/2). It is also worth to note that a Hadamard matrix of order n + 1 defines a tight 2-design in F n 2 , which is an (1, 1)-designs with cardinality n + 1 [12, Theorem 7.5]; i.e., exceeding our bound by 1. The divisibility condition now shows that this is the minimum possible cardinality for length n ≡ 3 (mod 4). Further examples of (1, 1)-designs can be extracted from the examples in [13] and [5], where linear programming bounds for codes with given minimum and maximum distances are considered.
The classification of tight 4-designs was recently completed by Gavrilyuk, Suda and Vidali [11] (see also [18]). The only tight 4-design is the unique even-weight code of length 5 (see Example 2.5). It has cardinality 16, which is the minimum possibility for a (2, 2)-design of length 5 since in this case our bound is 11 and the cardinality must be divisible by 2 4 = 16. i.e. n(n 2 − 3n + 8) is divisible by 2 7 . This gives n ≡ 0 (mod 8) or n ≡ 107 (mod 128).
Since Q 1,1 3 (t) has roots 0 and ± √ 3n − 8/n (the later necessarily belonging to T n ; otherwise C would be an equidistant code with the only allowed distance n/2), it follows that 3n − 8 is a perfect square. Setting n = 8u and 3n − 8 = m 2 , we easily see that u has to be odd and m cannot be multiple of 3. If n ≡ 107 (mod 128), we obtain m ≡ 43 (mod 64).
The necessary conditions are fulfilled for n = 24, where the Golay code, which is a tight 7-design, produces as in Theorem 2.3 a tight (3, 3)-design of 2 11 = 2048 points.