Semidefinite bounds for nonbinary codes based on quadruples

• Published in: Designs, codes, and cryptography Vol. 84; no. 1-2; pp. 87 - 100
• Main Authors: Litjens, Bart, Polak, Sven, Schrijver, Alexander
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2017; Springer Nature B.V
• Subjects: Circuits; Codes; Coding and Information Theory; Computer Science; Cryptology; Data Structures and Information Theory; Discrete Mathematics in Computer Science; Information and Communication; Integers; Mathematical analysis; Matrix methods; Semidefinite programming; Upper bounds; Semidefinite programming; 05E10; Upper bounds; 20C30; 90C22; Delsarte; Code; 94B65; Nonbinary code
• ISSN: 0925-1022; 1573-7586; 1573-7586
• DOI: 10.1007/s10623-016-0216-5

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 C k be the collection of codes of cardinality at most k . Then A q ( n , d ) is at most the maximum value of ∑ v ∈ [ q ] n x ( { v } ) , where x is a function C 4 → R + such that x ( ∅ ) = 1 and x ( C ) = 0 if C has minimum distance less than d , and such that the C 2 × C 2 matrix ( x ( C ∪ C ′ ) ) C , C ′ ∈ C 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 ) ≤ 176 , A 4 ( 7 , 3 ) ≤ 596 , A 4 ( 7 , 4 ) ≤ 155 , A 5 ( 7 , 4 ) ≤ 489 , and A 5 ( 7 , 5 ) ≤ 87 .