The Support Designs of Several Families of Lifted Linear Codes

TL;DR

This paper develops fundamental theory for lifted linear codes, analyzes their support designs, and determines weight distributions for certain lifted Reed-Muller codes, revealing new combinatorial structures.

Contribution

It introduces new theoretical results on lifted linear codes, including support design properties and weight distributions, expanding understanding of their combinatorial and algebraic structures.

Findings

Support 2-designs for lifted projective Reed-Muller, Hamming, and Simplex codes

Weight distributions of lifted Reed-Muller codes of certain orders

Infinite family of three-weight projective codes over GF(4)

Abstract

A generator matrix of a linear code $\C$ over $\gf(q)$ is also a matrix of the same rank $k$ over any extension field $\gf(q^\ell)$ and generates a linear code of the same length, same dimension and same minimum distance over $\gf(q^\ell)$, denoted by $\C(q|q^\ell)$ and called a lifted code of $\C$. Although $\C$ and their lifted codes $\C(q|q^\ell)$ have the same parameters, they have different weight distributions and different applications. Few results about lifted linear codes are known in the literature. This paper proves some fundamental theory for lifted linear codes, and studies the support $2$-designs of the lifted projective Reed-Muller codes, lifted Hamming codes and lifted Simplex codes. In addition, this paper settles the weight distributions of the lifted Reed-Muller codes of certain orders, and investigates the support $3$-designs of these lifted codes. As a by-product, an infinite family of three-weight projective codes over $\gf(4)$ is obtained.