Characteristic vector and weight distribution of a linear code

TL;DR

This paper introduces an algorithm to compute the weight distribution of linear codes over finite fields using characteristic vectors and a specific generator matrix, with complexity O(kq^k).

Contribution

It presents a novel algorithm leveraging characteristic vectors and specialized generator matrices to efficiently determine weight distributions of linear codes.

Findings

Algorithm computes weight distribution with complexity O(kq^k)

Uses characteristic vectors relative to generator matrices

Applicable to linear codes over finite fields

Abstract

We develop an algorithm for computing the weight distribution of a linear $[n,k]$ code over a finite field $\mathbb{F}_q$. We represent the codes by their characteristic vector with respect to a given generator matrix and a special type of a generator matrix of the k-dimensional simplex code. This characteristic vector is the input data of our algorithms. The complexity of the presented algorithms is O(kq^k).