A new upper bound on the minimal distance of self-dual codes

1990/01/01 by J.H. Conway, John H. Conway, N. J. A. Sloane +1 · 13 citations
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #graph theory and CDMA systems #Finite Group Theory Research

paper · doi:10.1109/18.59931

Abstract

It is shown that the minimal distance d of a binary self-dual code of length n>or=74 is at most 2((n+6)/10). This bound is a consequence of some new conditions on the weight enumerator of a self-dual code obtained by considering a particular translate of the code, called its shadow. These conditions also enable one to find the highest possible minimal distance of a self-dual code for all n>or=60; to show that self-dual codes with d<or=6 exist precisely for n>or=22, with d>or=8 exist precisely for n=24, 32 and n>or=26, and with d>or=10 exist precisely for n>or=46; and to show that there are exactly eight self-dual codes of length 32 with d=8. Several of the self-dual codes of length 34 have trivial group (this appears to be the smallest length where this can happen).<<ETX>>

Citations

• The children of the (32, 16) doubly even codes

Cited by

• Shadow bounds for self-dual codes
• New asymptotic bounds for self-dual codes and lattices
• Construction of MDS self-dual codes from orthogonal matrices
• New binary self-dual codes via a generalization of the four circulant\n construction
• The automorphism group of the doubly-even [72,36,16] code can only be of order 1, 3 or 5
• A group induced four-circulant construction for self-dual codes and new extremal binary self-dual codes
• On the Existence of Certain Optimal Self-Dual Codes with Lengths Between 74 and 116
• New extremal binary self-dual codes of lengths 66 and 68 from codes over r_k,m
• Note on the residue codes of self-dual ℤ₄-codes having large minimum Lee weights
• New Self-Dual Codes from 2x2 Block Circulant Matrices, Group Rings and Neighbours of Neighbours
• Convolutional encoding of 60,64,68,72-bit self-dual codes
• New Self-Dual Codes of length 68 from a 2 by 2 block matrix Construction and Group Rings
• On extremal double circulant self-dual codes of lengths 90-96

Related

• Shadow bounds for self-dual codes
• New Extremal binary self-dual codes of length 68 from a novel approach to neighbors
• Ternary near-extremal self-dual codes of lengths 36, 48 and 60
• On involutions in extremal self-dual codes and the dual distance of semi self-dual codes
• New Binary Self-Dual Cyclic Codes with Square-Root-Like Minimum Distances