Abstract
For n,d,w in mathbb {N}, let A(n, d, w) denote the maximum size of a binary code of word length n, minimum distance d and constant weight w. Schrijver recently showed using semidefinite programming that A(23,8,11)=1288, and the second author that A(22,8,11)=672 and A(22,8,10)=616. Here we show uniqueness of the codes achieving these bounds. Let A(n, d) denote the maximum size of a binary code of word length n and minimum distance d. Gijswijt et al. showed that A(20,8)=256. We show that there are several nonisomorphic codes achieving this bound, and classify all such codes with all distances divisible by 4.
Highlights
Let F2 := {0, 1} denote the field of two elements and fix n ∈ N
For any code C, the minimum distance dmin(C) (∈ R∪{∞}) of C is the minimum distance between any pair of distinct code words in C
We show using the output of the corresponding semidefinite programs that the codes of maximum size are unique for these n, d, w
Summary
Let F2 := {0, 1} denote the field of two elements and fix n ∈ N. A(n, d, w) is defined as the maximum size of a binary constant weight w code of minimum distance at least d. For unrestricted (non-constant weight) binary codes, the bound A(n, d) = A(20, 8) ≤ 256 was obtained in [7], implying that the quadruply shortened extended binary Golay code of size 256 is optimal. The quadruply shortened extended binary Golay code is a linear (n, d) = (20, 8)-code of size 256 and has all distances divisible by 4. It turns out that the 4 times shortened extended binary Golay code is not the only (20, 8)-code of size 256. We classify such codes with all distances divisible by 4, and find 15 such codes
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
