Derivatives of Complete Weight Enumerators and New Balance Principle of Binary Self-Dual Codes

TL;DR

This paper introduces derivatives of complete weight enumerators for binary self-dual codes, revealing new balance principles and applying them to classify and analyze specific codes like Golay and quadratic residue codes.

Contribution

It defines derivatives of weight enumerators that reduce code information while preserving eigenvector properties, leading to new balance equations for code analysis.

Findings

Computed derivatives for key codes like Golay and quadratic residue codes.

Established a new balance equation involving code vectors with fixed coordinate bits.

Applied the balance equation to eliminate possible weight enumerators for length-eight codes.

Abstract

Let H be the standard Hadamard matrix of order two and let K=2^{-1/2}H. It is known that the complete weight enumerator $\ W$ of a binary self-dual code of length $n$ is an eigenvector corresponding to an eigenvalue 1 of the Kronecker power $K^{[n]}.$ For every integer $t$ in the interval [0,n] we define the derivative of order $t$, $W_{<t>},$ of $W$ in such a way that $W_{<t>}$ is in the eigenspace of $\ 1$ of the matrix $K^{[n-t]}.$ For large values of $t,$ $W_{<t>}$ contains less information about the code but has smaller length while $W_{<0>}=W$ completely determines the code. We compute the derivative of order $n-5$ for the extended Golay code of length 24, the extended quadratic residue code of length 48, and the putative [72,24,12] code and show that they are in the eigenspace of $\ 1$ of the matrix $% K^{[5]}.$ We use the derivatives to prove a new balance equation which involves the number of code vectors of given weight having 1 in a selected coordinate position. As an example, we use the balance equation to eliminate some candidates for weight enumerators of binary self-dual codes of length eight.