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Quasi-optimal cyclic orbit codes

Chiara Castello, Heide Gluesing-Luerssen, Olga Polverino, Ferdinando Zullo

TL;DR

The paper develops a framework linking cyclic orbit codes to $\mathbb{F}_q$-linear sets on the projective line, enabling refined invariants under equivalence and sharper bounds on code parameters. It proves a general existence theorem for quasi-optimal full-length codes in even dimensions, and introduces explicit constructions via $U_{s,\gamma}$ that are full-length with a clear norm-based dichotomy between optimal and quasi-optimal cases. The authors also classify automorphism and Frobenius-automorphism structures for these codes and establish when two codes are equivalent under semilinear or Frobenius isometries, enriching the theory of code equivalence and symmetry. Together, these results advance explicit quasi-optimal code design, provide tightened bounds through linear-set arguments, and offer a comprehensive treatment of the symmetries and invariants of cyclic orbit codes.

Abstract

We focus on two aspects of cyclic orbit codes: invariants under equivalence and quasi-optimality. Regarding the first aspect, we establish a connection between the codewords of a cyclic orbit code and a certain linear set on the projective line. This allows us to derive new bounds on the parameters of the code. In the second part, we study a particular family of (quasi-)optimal cyclic orbit codes and derive a general existence theorem for quasi-optimal codes in even-dimensional vector spaces over finite fields of any characteristic. Finally, for our particular code family we describe the automorphism groups under the general linear group and a suitable Galois group.

Quasi-optimal cyclic orbit codes

TL;DR

The paper develops a framework linking cyclic orbit codes to -linear sets on the projective line, enabling refined invariants under equivalence and sharper bounds on code parameters. It proves a general existence theorem for quasi-optimal full-length codes in even dimensions, and introduces explicit constructions via that are full-length with a clear norm-based dichotomy between optimal and quasi-optimal cases. The authors also classify automorphism and Frobenius-automorphism structures for these codes and establish when two codes are equivalent under semilinear or Frobenius isometries, enriching the theory of code equivalence and symmetry. Together, these results advance explicit quasi-optimal code design, provide tightened bounds through linear-set arguments, and offer a comprehensive treatment of the symmetries and invariants of cyclic orbit codes.

Abstract

We focus on two aspects of cyclic orbit codes: invariants under equivalence and quasi-optimality. Regarding the first aspect, we establish a connection between the codewords of a cyclic orbit code and a certain linear set on the projective line. This allows us to derive new bounds on the parameters of the code. In the second part, we study a particular family of (quasi-)optimal cyclic orbit codes and derive a general existence theorem for quasi-optimal codes in even-dimensional vector spaces over finite fields of any characteristic. Finally, for our particular code family we describe the automorphism groups under the general linear group and a suitable Galois group.
Paper Structure (10 sections, 34 theorems, 92 equations)

This paper contains 10 sections, 34 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Linear sets
  3. Cyclic orbit codes
  4. A bound on $k$ for cyclic orbit codes with prescribed minimum distance
  5. Fundamental properties of the intersection distribution
  6. New bounds on $f_U$
  7. Constructions of quasi-optimal codes
  8. Semilinear Isometries and Frobenius Isometries
  9. Automorphisms group of $\mathrm{Orb}(U_{s,\gamma})$
  10. Open Problems

Key Result

Theorem 2.1

An $\mathbb{F}_{q}$-linear set $L_W$ of $W$-rank $k\leq n$ in $\mathrm{PG}(1,q^n)$ with at least one point of $W$-weight one has at least $q^{k-1}+1$ points.

Theorems & Definitions (76)

  • Theorem 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Theorem 3.1
  • Definition 3.2
  • Definition 3.3: see also heideweight
  • Theorem 3.4
  • Proposition 4.1
  • ...and 66 more