Bounds on codes over an alphabet of five elements

AbstractWe consider the problem of finding bounds and exact values of A5(n,d) — the maximum size of a code of length n and minimum distance d over an alphabet of 5 elements. Using a wide variety of constructions and methods, a table of bounds on A5(n,d) for n⩽11 is obtained

Semidefinite bounds for nonbinary codes based on quadruples

For nonnegative integers $q,n,d$, let $A_q(n,d)$ 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 $A_q(n,d)$. For any $k$, let \CC_k be the collection of codes of cardinality at most $k$. Then $A_q(n,d)$ is at most the maximum value of $\sum_{v\in[q]^n}x(\{v\})$, where $x$ is a function \CC_4\to R_+ such that $x(\emptyset)=1$ and $x(C)=0$ if $C$ has minimum distance less than $d$, and such that the \CC_2\times\CC_2 matrix (x(C\cup C'))_{C,C'\in\CC_2} 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 $A_4(6,3)\leq 176$, $A_4(7,4)\leq 155$, $A_5(7,4)\leq 489$, and $A_5(7,5)\leq 87$

New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming

We give a new upper bound on the maximum size $A_q(n,d)$ of a code of word length $n$ and minimum Hamming distance at least $d$ over the alphabet of $q\geq 3$ letters. By block-diagonalizing the Terwilliger algebra of the nonbinary Hamming scheme, the bound can be calculated in time polynomial in $n$ using semidefinite programming. For $q=3,4,5$ this gives several improved upper bounds for concrete values of $n$ and $d$. This work builds upon previous results of A. Schrijver [IEEE Trans. Inform. Theory 51 (2005), no. 8, 2859--2866] on the Terwilliger algebra of the binary Hamming schem

On independent star sets in finite graphs

Let G be a finite graph with μ as an eigenvalue of multiplicity k. A star set for μ is a set X of k vertices in G such that μ is not an eigenvalue of G-X. We investigate independent star sets of largest possible size in a variety of situations. We note connections with symmetric designs, codes, strongly regular graphs, and graphs with least eigenvalue -2

Semidefinite bounds for nonbinary codes based on quadruples

For nonnegative integers q, n, d, let 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 Aq(n, d). For any k, let Ck be the collection of codes of cardinality at most k. Then Aq(n, d) is at most the maximum value of Pv∈[q]n x({v}), where x is a function C4 → R+ such that x(∅) = 1 and x(C) = 0 if C has minimum distance less than d, and such that the C2 ×C2 matrix (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 A4(6, 3) ≤ 176, A4(7, 4) ≤ 155, A5(7, 4) ≤ 489, and A5(7, 5) ≤ 87