Linear Codes from Some 2-Designs

TL;DR

This paper introduces new linear codes derived from specific 2-designs, including almost difference sets and semibent function-related designs, resulting in optimal codes with applications in security and data systems.

Contribution

It presents a novel construction method for linear codes using particular 2-designs, expanding beyond classical incidence matrix approaches.

Findings

Two families of codes are optimal.

Codes have few weights, beneficial for applications.

Applications include secret sharing and authentication.

Abstract

A classical method of constructing a linear code over $\gf(q)$ with a $t$-design is to use the incidence matrix of the $t$-design as a generator matrix over $\gf(q)$ of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of $2$-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of $2$-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterisation of highly nonlinear Boolean functions is presented.