Semidefinite bounds for nonbinary codes based on quadruples

For nonnegative integers q, n, d, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_q(n,d)$$\end{document}Aq(n,d) denote the maximum cardinality of a code of length n over an alphabet [q] with q letters and with minimum distance at least d. We consider the following upper bound on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_q(n,d)$$\end{document}Aq(n,d). For any k, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{C}_k$$\end{document}Ck be the collection of codes of cardinality at most k. Then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_q(n,d)$$\end{document}Aq(n,d) is at most the maximum value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{v\in [q]^n}x(\{v\})$$\end{document}∑v∈[q]nx({v}), where x is a function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{C}_4\rightarrow {\mathbb {R}}_+$$\end{document}C4→R+ such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x(\emptyset )=1$$\end{document}x(∅)=1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x(C)=\!0$$\end{document}x(C)=0 if C has minimum distance less than d, and such that the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{C}_2\times \mathcal{C}_2$$\end{document}C2×C2 matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x(C\cup C'))_{C,C'\in \mathcal{C}_2}$$\end{document}(x(C∪C′))C,C′∈C2 is positive semidefinite. By the symmetry of the problem, we can apply representation theory to reduce the problem to a semidefinite programming problem with order bounded by a polynomial in n. It yields the new upper bounds \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_4(6,3)\le 176$$\end{document}A4(6,3)≤176, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_4(7,3)\le 596$$\end{document}A4(7,3)≤596, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_4(7,4)\le 155$$\end{document}A4(7,4)≤155, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_5(7,4)\le 489$$\end{document}A5(7,4)≤489, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_5(7,5)\le 87$$\end{document}A5(7,5)≤87.


Introduction
Let Z + denote the set of nonnegative integers, and denote [m] = {1, . . ., m}, for any m ∈ Z + .Fixing n, q ∈ Z + , a code is a subset of [q] n .So [q] serves as the alphabet and n as the word length.We will assume throughout that q ≥ 2. (If you prefer {0, 1, . . ., q − 1} as alphabet, take the letters mod q.)While this paper is mainly meant to handle the case q ≥ 3, the results also hold for q = 2.
For v, w ∈ [q] n , the (Hamming) distance d H (v, w) is equal to the number of i ∈ [n] with v i = w i .The minimum distance of a code C is the minimum of d H (v, w) taken over distinct v, w ∈ C. Then A q (n, d) denotes the maximum cardinality of a code with minimum distance at least d.We will study the following upper bound on A q (n, d), sharpening Delsarte's classical linear programming bound [4].
For k ∈ Z + , let C k be the collection of subsets C of [q] n with |C| ≤ k.For each x : C 4 → R define the C 2 × C 2 matrix M (x) by ( 1) x({w}), where x : C 4 → R + satisfies (i) x(∅) = 1, (ii) x(C) = 0 if the minimum distance of C is less than d, (iii) M (x) is positive semidefinite.
Proof.Let D ⊆ [q] n have minimum distance at least d and satisfy |D| = A q (n, d).Define x : C 4 → R by x(C) = 1 if C ⊆ D and x(C) = 0 otherwise.Then x satisfies the conditions: (iii) follows from the fact that for this x one has M (x) C,C ′ = x(C)x(C ′ ) for all C, C ′ ∈ C 2 .Moreover, w∈[q] n x({w}) = |D| = A q (n, d).
The optimization problem (2) is huge, but, with methods from representation theory, can be reduced to a size bounded by a polynomial in n, with entries (i.e., coefficients) being polynomials in q.This makes it possible to solve (2) by semidefinite programming for some moderate values of n, d, and q, leading to improvements of best known upper bounds for A q (n, d).
To explain the reduction, let H be the wreath product S n q ⋊ S n .For each k, the group H acts naturally on C k , maintaining minimum distances and cardinalities of elements of C k (being codes).Then we can assume that x is invariant under the H-action on C 4 .That is, we can assume that x(C) = x(D) whenever C, D ∈ C 2 and D = g•C for some g ∈ H. Indeed, (2)(i)(ii)(iii) are maintained under replacing x by g • x. (Note that M (g • x) is obtained from M (x) by simultaneously permuting rows and columns.)Moreover, the objective function does not change by this action.Hence the optimum x can be replaced by the average of all g • x (over all g ∈ H), by the convexity of the set of positive semidefinite matrices.This makes the optimum solution H-invariant.
Let Ω be the set of H-orbits on C 4 .Note that Ω is bounded by a polynomial in n (independently of q).As there exists an H-invariant optimum solution, we can replace, for each ω ∈ Ω and C ∈ ω, each variable x(C) by a variable y(ω).In this way we obtain M (y).
Then M (y) is invariant under the action of H on its rows and columns, induced from the action of H on C 2 .Hence M (y) can be block-diagonalized by M (y) → U T M (y)U , where U is a matrix independent of y.The entries in each block are linear functions of the variables y(ω).There are several equal (or equivalent) blocks.Taking one block from each such class gives a matrix of order polynomial in n with numbers that are polynomials in q.The issue crucial for us is that the original matrix M (y) is positive semidefinite if and only if each of the blocks is positive semidefinite.
In this paper we will describe the blocks that reduce the problem.With this, we found the following improvements on the known bounds for A q (n, d), with thanks to Hans D. The best upper bound 179 previously known for A 4 (6, 3) is the Delsarte bound [4]; the other three best upper bounds previously known were given by Gijswijt, Schrijver, and Tanaka [7].We refer to the most invaluable tables maintained by Andries Brouwer [3] with the best known lower and upper bounds for the size of error-correcting codes (see also Bogdanova, Brouwer, Kapralov, and Östergård [1] and Bogdanova and Östergård [2] for studies of bounds for codes over alphabets of size q = 4 and q = 5, respectively).

Comparison with earlier bounds
The bound B q (n, d) described above is a sharpening of Delsarte's classical linear programming bound [4].The value of the Delsarte bound is equal to our bound after replacing C 4 and C 2 by C 2 and C 1 , respectively, which generally yields a less strict bound.We can add to (2) the condition that, for each D ∈ C 4 , the S(D) × S(D) matrix ) Also the addition of (3) allows a reduction of the optimization problem to polynomial size as above.For q = 2 we obtain in this way the bound given by Gijswijt, Mittelmann, and Schrijver [6].Our present description gives a more conceptual and representationtheoretic approach to the method of [6].Some preliminary computations suggest that adding condition (3) might be superfluous in the optimization problem, but this needs further investigation.
A bound intermediate to the Delsarte bound and the currently investigated bound is based on considering functions x : C 3 → R + and the related matrices -see Schrijver [9] for binary codes and Gijswijt, Schrijver, and Tanaka [7] for nonbinary codes.

Preliminaries on representation theory
We assume some familiarity with classical representation theory, in particular of the symmetric group S n and of finite groups in general.In this section we give a brief review, also to settle some notation and terminology.We refer to Sagan [8] for background.All groups considered are finite, which allows us to keep decompositions and reductions real-valued.
A group G acts on a set X if there is a group homomorphism G → S X , where S X is the set of bijections X → X.The image of g ∈ G in S X is indicated by g • .If X is a linear space, the bijections are assumed to be linear functions.The action of G on a set X induces an action of G on the linear space R X , by (g If any group G acts on X, then X G denotes the set of elements of X invariant under the action of G.
Let m ∈ Z + and let G be a finite group acting on V = R m .Then V can be decomposed uniquely as direct sum of the (This decomposition is generally not unique.)For each i ≤ k and j ≤ m i , choose a nonzero u i,j ∈ V i,j such that for each i and all j, j ′ ≤ m i there exists a G-isomorphism V i,j → V i,j ′ bringing u i,j to u i,j ′ .For each i ≤ k, let U i be the matrix [u i,1 , . . ., u i,m i ], considering the u i,j as columns.We call any matrix set {U 1 , . . ., U k } that can be obtained in this way representative for the action of G on R m .It has the property that the function i is equal to the dimension of (R m×m ) G (and hence can be considerably smaller than m).Another important property of a representative matrix set is that X is positive semidefinite if and only if Φ(X) is positive semidefinite.Moreover, by construction, as V i,j = RG • u i,j for all i, j, It will turn out to be convenient to consider the columns of the matrices U i as elements of the dual space (R m ) * (by taking the standard inner product).Then each U i is an ordered set of linear functions on R m .(The order plays a role in describing a representative matrix set for the action of the wreath product G ⋊ S n on V ⊗n .)

A representative set for the action of S n on V ⊗n
Classical representation theory of the symmetric group yields a representative set for the natural action of S n on V ⊗n , where V is a finite-dimensional vector space, which we will describe now. For For any j 0 ≤ t, the set of elements (i, j 0 ) in Y (λ) is called the j 0 -th row of Y (λ).Let R λ be the group of permutations π of Y (λ) with π(Z) = Z for each row Z of Y (λ).For any Let B = (B(1), . . ., B(m)) be an ordered basis of V * .For τ ∈ T λ,m , define the following element of (V * ) ⊗n : (7) u τ,B := where we order the Young shape Y (λ) by concatenating its rows.Then the matrix set is representative for the natural action of S n on V ⊗n .

Reduction of the optimization problem
In this section we describe reducing the optimization problem (2) conceptually.In Section 4 we consider the reduction computationally.For the remainder of this paper we fix n and q.
We consider the natural action of H = S q ⋊S n on R C 2 .If U 1 , . . ., U k form a representative set of matrices for this action, then with (4) we obtain a reduction of the size of the optimization problem to polynomial size.To make this reduction explicit in order to apply semidefinite programming, we need to express each m i ×m i matrix U T i M (y)U i as an explicit matrix in which each entry is a linear combination of the variables y(ω) for ω ∈ Ω (the set of H-orbits of C 4 ).
For ω ∈ Ω, let N ω be the C 2 × C 2 matrix with 0, 1 entries satisfying So to get the reduction, we need to obtain the matrices U T i N ω U i explicitly, for each ω ∈ Ω and for each i = 1, . . ., k.We do this in a number of steps.
We first describe in Section 3.1 a representative set for the natural action of S q on R q×q .From this we derive, in Section 3.2, with the help of the representative set for the action of S n on V ⊗n described in Section 2.1, a representative set for the action of the wreath product H = S n q ⋊ S n on the set ([q] n ) 2 of ordered pairs of words in [q] n , in other words, on R ([q] n ) 2 ∼ = (R q×q ) ⊗n .From this we derive in Section 3. ) Finally, in Section 3.4 we include the empty set ∅, by an easy representation-theoretic argument.

A representative set for the action of S q on R q×q
We now consider the natural action of S q on R q×q .Let e j be the j-th unit basis vector in R q , J q be the all-one q × q matrix, 1 be the all-one column vector in R q , N := (e 1 − e 2 )1 T , and E i,j := e i e T j .We furthermore define the following matrices, where we consider matrics in R q×q as columns of the matrices B i : The matrices in R q×q will in fact be taken as elements of the dual space (R q×q ) * (by taking the inner product), so that they are elements of the algebra O(R q×q ) of polynomials on the linear space R q×q .
For the remainder of this paper we fix k, m 1 , . . ., m k , and B 1 , . . ., B k .

A representative set for the action of H on (R q×q ) ⊗n
We next consider the action of H on the set ([q] n ) 2 of ordered pairs of code words.For that, we derive a representative set for the natural action of H on (R q×q ) ⊗n ∼ = R ([q] n ) 2 from the results described in Sections 2.1 and 3.1.

Unordered pairs
We now go over from the set ([q] n ) 2 of ordered pairs of code words to the set C 2 \ {∅} of unordered pairs (including singletons) of code words.For this we consider the action of the group S 2 on R , where the nonidentity element σ in S 2 acts as taking the transpose.The actions of S 2 and H commute.

Restriction to pairs of words at distance at least d
Let d ∈ Z + , and let C d 2 be the collection of elements of C 2 of minimum distance at least d.Note that each singleton code word belongs to C d 2 , and that H acts on C d 2 .From (20) we derive a representative set for the action of H on R C d 2 \{∅} .To see this, let for each t ∈ Z + , L t be the subspace of R C 2 spanned by the elements e {α,β} with α, β ∈ [q] n and d H (α, β) = t.(For any Z ∈ C 2 , e Z denotes the unit base vector in R C d 2 for coordinate Z.) Then for any n ∈ N , λ ⊢ n, and τ ∈ W ′ λ , the irreducible representation are the only two entries B i (j) in the B i that have nonzeros on the diagonal of the matrix B i (j).Let W ′′ λ be the set of those τ in Then a representative set for the action of

Adding ∅
To obtain a representative set for the action of H on C d 2 , note that H acts trivially on ∅.So e ∅ belongs to the H-isotopical component of R C 2 that consists of H-invariant elements.Now the H-isotypical component of R C 2 \{∅} that consists of the H-invariant elements corresponds to the matrix in the representative set indexed by indexed by n = (n, 0, 0, 0) and λ = ((n), (), (), ()), where () ⊢ 0. So to obtain a representative set for R C 2 , we just add e ∅ as column to this matrix.

How to compute (
We now have a reduction of the original problem to blocks with coefficients (F v τ ) T N ω F v σ , for n ∈ N , λ ⊢ n, τ , σ ∈ W λ , and ω ∈ Ω.The number and orders of these blocks are bounded by a polynomial in n, but computing these coefficients still must be reduced in time, since the order of F , v τ , v σ , and N ω is exponential in n.
Let Π be the collection of partitions of {1, 2, 3, 4} into at most q parts.There is the following bijection between Π and the set of orbits of the action of S q on [q] 4 .
For each word w ∈ [q] 4 , let part(w) be the partition P ∈ Π such that i and j belong to the same class of P if and only if w i = w j (for i, j = 1, . . ., 4).Then two elements v, w ∈ [q] 4 belong to the same S q -orbit if and only if part(v) = part(w).Note that |Π| = 8 if q = 2, |Π| = 14 if q = 3, and |Π| = 15 if q ≥ 4. (In all cases, |Π| = dim(R q×q ) Sq = k i=1 m 2 i .)For P ∈ Π, let (26) where each e i is a unit basis column vector in R q , so that e i e T j is a matrix in R q×q .Then D := {d P | P ∈ Π} is a basis of (R q×q ⊗ R q×q ) Sq .Let D * be the dual basis.
By this lemma, it suffices to compute v T τ K µ v σ for each µ ∈ Q.To this end, define the following degree n polynomial on W := (R q×q ⊗ R q×q ) Sq : This polynomial can be computed (i.e., expressed as linear combination of monomials in B i (j)⊗B i (h)) in time bounded by a polynomial in n (Gijswijt [5], see Appendix 1 in Section where ω is the H-orbit of C 4 consisting of all pairs {α, β} with d H (α, β) = t.

Appendix 1: Computation of p τ,σ
For any n, m ∈ Z + , λ ⊢ n, and τ, σ ∈ T λ,m , define the polynomial Proposition 3. Expressing p τ,σ as a linear combination of monomials can be done in polynomial time, for fixed m.
Proof.First observe that (λ * is the dual partition of λ; that is, λ * j is the height of column j.)For fixed m, when n grows, there will be several columns of Y (λ) that are the same both in τ ′ and in σ ′ .More precisely, for given τ ′ , σ ′ let the 'count function' κ be defined as follows: for t ∈ Z + and v, w ∈ [m] t , κ(v, w) is the number of columns j of height t such that τ ′ (i, j) = v i and σ ′ (i, j) = w i for all i = 1, . . ., t.Then for each i ≤ h := height(λ) and each s ∈ [m]:

Appendix 2: Expressing
Recall that each B i (j) is a linear function on R q×q , and that each d P is an element of R q×q ⊗ R q×q , where P belongs to the set Π of partitions of {1, . . ., 4} with at most q classes.We express each B i (j)⊗B i (h) in the dual basis B * := {d * P | P ∈ Π}.The coefficient of d * P is obtained by evaluating (B i (j) ⊗ B i (h))(d P ).This is routine, but we display the expressions.