Primal-dual distance bounds of linear codes with application to cryptography

TL;DR

This paper establishes bounds on the minimum length of linear codes with given minimum distances and applies these results to cryptographic Boolean function design.

Contribution

It provides new lower and upper bounds for the length of linear codes with specified distances and determines optimal codes for small parameters.

Findings

Derived bounds for N(d,d^⊥)

Determined optimal codes for small d and d^⊥

Connected code bounds to cryptographic Boolean function design

Abstract

Let $N(d,d^\perp)$ denote the minimum length $n$ of a linear code $C$ with $d$ and $d^{\bot}$, where $d$ is the minimum Hamming distance of $C$ and $d^{\bot}$ is the minimum Hamming distance of $C^{\bot}$. In this paper, we show a lower bound and an upper bound on $N(d,d^\perp)$. Further, for small values of $d$ and $d^\perp$, we determine $N(d,d^\perp)$ and give a generator matrix of the optimum linear code. This problem is directly related to the design method of cryptographic Boolean functions suggested by Kurosawa et al.