УДК 53 B. Mounits NEW UPPER BOUNDS FOR NONBINARY CODES

New upper bounds on codes are presented. The bounds are obtained by linear and semidefinite programming. INTRODUCTION One of the central problems in coding theory is to find upper bounds on maximum size Aq(n, d) of a code of word length n and minimum Hamming distance at least d over the alphabet Q of q ≥ 2 letters. Let us provide Q with the structure of an Abelian group, in an arbitrary way. In 1973 Delsarte proposed a linear programming approach for bounding the size of cliques in an association scheme. This bound is based on diagonalizing the BoseMesner algebra of the scheme. To obtain bounds on Aq(n, d), Delsarte introduced the Hamming scheme H(n,q), which is generated by action of a group of permutations of Q that preserve the Hamming distance. In 2005 Schrijver gave a new upper bound on A2(n,d) using semidefinite programming, which is obtained by block-diagonalizing the ( ) – dimensional Terwilliger algebra of H(n,2). The semidefinite programming bound for Aq(n,d), based on the block-diagonalizing the ( ) – dimensional Terwilliger algebra of H(n, q) was presented later by Gijswijt, Schrijver and Tanaka. In this work we introduce an association scheme which is generated by a subgroup of permutations of Q that preserve not only the Hamming distance, but also the "type" of the difference of vectors. The dimension of the Bose-Mesner algebra of this scheme is ( ). We also describe the ( ) – dimensional Terwilliger algebra of this new scheme. In particular, we have found that the orbits of Q x Q x Q under the action of the subgroup are characterized by certain q x q matrices. With these two algebras in hand, we derive a linear programming bound and a semidefinite programming bound for Aq(n, d) which generalize the bounds above. For the binary case, our scheme and the Hamming scheme H(n, 2) coincide. ASSOCIATION SCHEMES AND THE LP BOUND Let G –{g1 = 0, g2, ..., g|G|} denote an (additively written) arbitrary finite abelian group with zero element 0, and G = G x G x... x G denote an abelian group with respect to componentwise sum. For an integer n we denote Nn: = , ... , | | : ∈ {0,1, ... , }, ∑ = } ∈ . Define a function ψ: G→Nn as follows: ψ{х) := (cg1{x),cg2{x),..., cg|G|(x)), cg(x)=|{i:xi = g}|. A nonempty subset С of G is called a code of length n. For a set S ⊆ we define AG(n, S) := max{|C| : С ⊆ G, ψ (y – x) ∈S ⋁ , у ∈ С} . For α∈ let Rα be a relation Rα := {(x, y) ∈ G x G : ψ (y x) = α} and denote R = {Rα} α ∈ Nn. Let H denote a group consisting of the permutations of G obtained by permuting the n coordinates followed by adding a word from G, i.e., H = {π (•) + υ: π ∈ Sn, υ ∈ G}. It is obvious that H acts transitively on G. H has a natural action on G x G given by h(x,y) := (hx, hy). The following lemma states that the orbitals {(hx, hy) : h∈H} form the relations of R. Lemma 2.1. For any a ∈ Nn and x,y ∈ G such that (x,y) ∈ Rα there holds Rα = = {(hx,hy) : h ∈ H}. Proof: Let x, у ∈ X be such that ψ (у – x) = α. Thus, for h = (•) + v ∈ H, hy – hx = (πy + υ) – (πx + υ) = π(y – x) and ψ (hy – hx) = ψ (у – x) (1) which implies that {(hx, hy) : h ∈ H} ⊆Rα. On the other hand, we have to show that for any ( , ) ∈ Rα there exists h ∈ H such that ( , ) = (hx, hy). One can see that exists h0 ∈ H such that h0x = 0 and h0y – uα, where α0 αgi αg׀G׀ uα = 0. . . .0 ... ... ... | | ... | |, namely h0(•) = π0(•) – π0x for some π0 ∈ SnSimilarly, there exists h1 ∈ H such that h1 = = 0 and h1 – uα , namely h1(•)=π1(•) – π1 for some π1 ∈ Sn. Thus, h(•) = h1 h0(•) = π1 π0(•) + – π1 π0(x) satisfies (hx,hy) = ( , ) which proves the required inclusion. Theorem 2.2. (G,R) is a commutative association scheme with ( | | | | ) relations. Proof: It is well known (see for example [1]) that the orbitals from a group action form relations of an association scheme. For (x, у) ∈ Rγ denote Z(x,y): = {z∈ G : (x,z) ∈ Rα, (z,y) ∈ Rβ}, (x,y) = {z∈G : (x,z) ∈ Rβ, (z,y) ∈ Rα}. Since z ∈ Z(x,y) ↔(-z) ∈ (-y,-x) we conclude that pα,β = |Z(x,y)| = | (-y,-x)|= pβ, α . Note that the number of relations is equal to the number of (|G| – 1) – tuples of nonnegative integers (αg2, ..., αg|G) such that αg2 + ... + αg|G| ≤ n. Let Dα denote the adjacency matrix of the relation Rα, i.e., (Dα)x,y = 1, ( , ) ∈ , 0, h . The matrices {Dα} α ∈ form a basis of a commutative | | | | – dimensional Bose-Mesner algebra AGn of the scheme (G, R). In general, (G, R) is a non-symmetric association scheme. For α ∈ Nn, the inverse R α = {(y, x) : (x, y) ∈ Rα} of the relation Rα is given by R α = R where α ∶= α , ... , α | | , α = α . (2) It's easy to see that the valency of the relation Rα (and of ) is υα = ( , ,..., ) = = , ,..., | | . Consider the association scheme (G,R), where R={R }, R = R ∪ R . This is symmetric association scheme. Note that = + are symmetric matrices. We denote by the Bose-Mesner algebra of {G, R) and by = { } ∈ the group of characters. The next theorem gives more details about the symmetric scheme. Theorem 2.3. The unitary matrix U which diagonalizes the AGn is given by ( ) , = | | | ( ) . The primitive idempotent α, α ∈ , is the matrix with (x, y) entry ( ) , = | | ∑ ∈ ( ) { , } ( ) . (3) The eigenvalues are given by ( ) = ( ) = ∑ ∈ ( ) { , } ( ) (4) where u ∈ G is any word with ψ(u) ∈ {α, }. For α = (α0, ..., αg|G|)∈ there holds ∈ ∗ = ( ,..., | |)∈ ⋯ | | ( ). Where (x) is the Krawtchouk polynomial of degree k. A. Association Scheme for G = Z3. Let us look at an example for G = Z3 = {0, 1, 2}. For convenience we will omit a0. = {α – (α1, α2) : α1 + α2 ≤ n}. Thus, the number of relations in a non-symmetric scheme ( , ) is |R| = = , and the number of relations in the symmetric scheme ( ,R) is | | = ( ) , , ( )( ) , . The polynomial ( , ) (α , ) = ∑ , , , × × / ( )( / ( ) + (1 − , ) / ( )). We list here few polynomials: