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The Support Designs of Several Families of Lifted Linear Codes

Cunsheng Ding, Zhonghua Sun, Qianqian Yan

TL;DR

This work develops a comprehensive theory of lifted linear codes and leverages it to construct new incidence designs. It formalizes the effects of lifting on code parameters and weight distributions, and uses automorphism groups and design-theoretic criteria to generate large families of 2- and 3-designs from lifted codes. Key contributions include weight-distribution results and explicit construction of infinite families of $2$-designs from lifted projective Reed–Muller, Simplex, and Hamming codes, as well as abundant $3$-designs arising from lifted binary Reed–Muller codes. The results broaden the toolkit for connecting coding theory and combinatorial design, and they produce several open problems, including determining precise design parameters ($\lambda_i$) and weight enumerators for lifted codes in general. The work also yields an infinite family of three-weight projective codes over ${\mathrm GF}(4)$ and showcases the rich interaction between finite geometry, automorphism groups, and lifting operations in code design.

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.

The Support Designs of Several Families of Lifted Linear Codes

TL;DR

This work develops a comprehensive theory of lifted linear codes and leverages it to construct new incidence designs. It formalizes the effects of lifting on code parameters and weight distributions, and uses automorphism groups and design-theoretic criteria to generate large families of 2- and 3-designs from lifted codes. Key contributions include weight-distribution results and explicit construction of infinite families of -designs from lifted projective Reed–Muller, Simplex, and Hamming codes, as well as abundant -designs arising from lifted binary Reed–Muller codes. The results broaden the toolkit for connecting coding theory and combinatorial design, and they produce several open problems, including determining precise design parameters () and weight enumerators for lifted codes in general. The work also yields an infinite family of three-weight projective codes over and showcases the rich interaction between finite geometry, automorphism groups, and lifting operations in code design.

Abstract

A generator matrix of a linear code over is also a matrix of the same rank over any extension field and generates a linear code of the same length, same dimension and same minimum distance over , denoted by and called a lifted code of . Although and their lifted codes 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 -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 -designs of these lifted codes. As a by-product, an infinite family of three-weight projective codes over is obtained.
Paper Structure (12 sections, 26 theorems, 54 equations)

This paper contains 12 sections, 26 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. The support designs of linear codes
  3. Lifted linear codes
  4. The idea of constructing new designs with lifted codes of a linear code
  5. Motivations and objectives of this paper
  6. Fundamental results of lifted linear codes
  7. The support $2$-designs of the lifted projective Reed-Muller codes
  8. The support $2$-designs of the lifted Simplex and Hamming codes
  9. The support designs of Hamming and Simplex codes over finite fields
  10. The support designs of lifted Simplex and Hamming codes
  11. Lifted Reed-Muller codes and their support designs
  12. Summary and concluding remarks

Key Result

Theorem 1

Let ${\mathcal{C}}$ be a $[v,k,d]_q$ code. Let $d^\perp$ denote the minimum distance of the dual code ${\mathcal{C}}^\perp$ of ${\mathcal{C}}$. Let $w$ be the largest integer satisfying $w \leq v$ and Define $w^\perp$ analogously using $d^\perp$. Let $(A_i)_{i=0}^v$ and $(A_i^\perp)_{i=0}^v$ denote the weight distribution of ${\mathcal{C}}$ and ${\mathcal{C}}^\perp$, respectively. Fix a positive

Theorems & Definitions (43)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • ...and 33 more