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Linear codes over $\frac{\mathbb{F}_q[u]}{\langle u^2 \rangle}$ with mixed-alphabet defining sets and their Gray images: Constructions of projective few-weight, distance-optimal and minimal codes

Leijo Jose, Lavanya G., Anuradha Sharma

TL;DR

This work advances the construction of linear codes over the mixed-alphabet ring $\\mathcal{R}=\\frac{\\mathbb{F}_q[u]}{\\langle u^2 \\rangle}\\times \\mathbb{F}_q$ via defining sets drawn from simplicial complexes, and analyzes their Lee weights through Gray images to produce new families of few-weight, distance-optimal, and minimal codes over $\\mathbb{F}_q$. It provides explicit parameter formulas, spanning-matrix techniques for arbitrary defining sets, and two projective code families with new parameters, including self-orthogonal instances for small $q$. The paper also links these constructions to practical applications in secret sharing and locally repairable codes, and derives duals yielding binary distance-optimal and dimension-optimal codes, as well as a quaternary 3-weight family with sum-of-weights properties. Furthermore, it explores spectral graph implications by connecting certain codes to strongly walk-regular graphs and detail the locality properties of the resulting projective codes. Overall, the results address open questions on constructing distance- and parameter-optimal codes over rings and their Gray images, expanding the catalog of codes with new weight structures and practical applications.

Abstract

Let $\mathcal{R}=\frac{\mathbb{F}_q[u]}{\langle u^2 \rangle}\times \mathbb{F}_q$ be the mixed alphabet ring. In this paper, we construct four infinite families of linear codes over the ring $\frac{\mathbb{F}_q[u]}{\langle u^2 \rangle}$ whose defining sets are certain nonempty subsets of $\mathcal{R}^m$ associated with three simplicial complexes of $\mathbb{F}_q^m,$ each possessing a single maximal element. We explicitly determine the parameters and Lee weight distributions of these codes. We also study their Gray images and obtain three infinite families of few weight, near Griesmer, distance optimal and minimal codes over $\mathbb{F}_q$ with new parameters. We also provide two constructions of infinite families of projective few weight codes over $\mathbb{F}_q$ with new parameters, and observe that these codes are self orthogonal for $q=2$ or $3.$ Additionally, we obtain two infinite families of binary distance optimal projective codes and an infinite family of dimension optimal projective codes over $\mathbb{F}_q$ with new parameters. Apart from this, we construct an infinite family of quaternary projective $3$-weight codes whose non zero Hamming weights sum to $\frac{9}{4}$ times the code length, which give rise to strongly walk regular graphs. As an application of our newly constructed minimal codes over $\mathbb{F}_q$, we examine the minimal access structures of Massey's secret sharing schemes based on their duals and determine the number of dictatorial participants in these schemes. Finally, we investigate the locality properties of our newly constructed projective codes.

Linear codes over $\frac{\mathbb{F}_q[u]}{\langle u^2 \rangle}$ with mixed-alphabet defining sets and their Gray images: Constructions of projective few-weight, distance-optimal and minimal codes

TL;DR

This work advances the construction of linear codes over the mixed-alphabet ring via defining sets drawn from simplicial complexes, and analyzes their Lee weights through Gray images to produce new families of few-weight, distance-optimal, and minimal codes over . It provides explicit parameter formulas, spanning-matrix techniques for arbitrary defining sets, and two projective code families with new parameters, including self-orthogonal instances for small . The paper also links these constructions to practical applications in secret sharing and locally repairable codes, and derives duals yielding binary distance-optimal and dimension-optimal codes, as well as a quaternary 3-weight family with sum-of-weights properties. Furthermore, it explores spectral graph implications by connecting certain codes to strongly walk-regular graphs and detail the locality properties of the resulting projective codes. Overall, the results address open questions on constructing distance- and parameter-optimal codes over rings and their Gray images, expanding the catalog of codes with new weight structures and practical applications.

Abstract

Let be the mixed alphabet ring. In this paper, we construct four infinite families of linear codes over the ring whose defining sets are certain nonempty subsets of associated with three simplicial complexes of each possessing a single maximal element. We explicitly determine the parameters and Lee weight distributions of these codes. We also study their Gray images and obtain three infinite families of few weight, near Griesmer, distance optimal and minimal codes over with new parameters. We also provide two constructions of infinite families of projective few weight codes over with new parameters, and observe that these codes are self orthogonal for or Additionally, we obtain two infinite families of binary distance optimal projective codes and an infinite family of dimension optimal projective codes over with new parameters. Apart from this, we construct an infinite family of quaternary projective -weight codes whose non zero Hamming weights sum to times the code length, which give rise to strongly walk regular graphs. As an application of our newly constructed minimal codes over , we examine the minimal access structures of Massey's secret sharing schemes based on their duals and determine the number of dictatorial participants in these schemes. Finally, we investigate the locality properties of our newly constructed projective codes.

Paper Structure

This paper contains 15 sections, 33 theorems, 60 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Some preliminaries
  3. Linear codes over finite fields
  4. A mixed-alphabet ring of a quasi-Galois ring and its residue field, and the associated Euclidean bilinear form
  5. Gray images of linear codes over the quasi-Galois ring $\frac{\mathbb{F}_q[u]}{\langle u^2 \rangle}$
  6. Four infinite families of linear codes over $\frac{\mathbb{F}_q[u]}{\langle u^2 \rangle}$
  7. Some basic lemmas
  8. Parameters and Lee weight distributions of the codes $\mathscr{C}_{\mathcal{S}_1},$ $\mathscr{C}_{\mathcal{S}_2},$ $\mathscr{C}_{\mathcal{S}_3}$ and $\mathscr{C}_{\mathcal{S}_4}$ over $\frac{\mathbb{F}_q[u]}{\langle u^2 \rangle}$
  9. Some constructions of projective few-weight codes over $\mathbb{F}_q$
  10. Some additional applications
  11. Minimal access structures of Massey's secret sharing schemes based on the duals of our minimal codes
  12. Construction of locally repairable codes with localities either $2$ or $3$
  13. Conclusion and future work
  14. Acknowledgements
  15. Appendix: Coset graphs and strongly walk-regular graphs

Key Result

Lemma 2.1

Ashikhmin1998 Let $\mathcal{C}$ be a linear code over $\mathbb{F}_q,$ and let $w_0$ and $w_\infty$ denote the minimum and maximum among the Hamming weights of non-zero codewords of $\mathcal{C},$ respectively. If $\frac{w_0}{w_\infty}>\frac{q-1}{q},$ then the code $\mathcal{C}$ is minimal.

Theorems & Definitions (70)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 4.1
  • Lemma 4.1
  • proof
  • Theorem 4.1
  • ...and 60 more