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An intrinsical description of group codes

José Joaquín Bernal, Ángel del Río, Juan Jacobo Simón

TL;DR

This work gives an intrinsic criterion to determine when a linear code is a group code, by analyzing the permutation automorphism group PAut(C) and locating regular subgroups within it. It shows that a code is a left G-code precisely when G corresponds to a regular, transitive subgroup of S_n contained in PAut(C), and a G-code when H also satisfies H ∪ C_{S_n}(H) ⊆ PAut(C). The authors apply this framework to 1-dimensional and Cauchy codes, deriving concrete structural results: (i) one-dimensional left G-codes require a block-structure with a normal subgroup leading to a cyclic quotient, (ii) any G-code with G = AB for abelian A,B is abelian, yet left G-codes can be non-abelian, and (iii) for q-ary Cauchy codes of length q or q−1 the possible left G-code structures are tightly controlled (cyclic or dihedral) with precise congruence conditions; in particular, many large-location codes reduce to cyclic forms when q ≡ −1 (mod 4). Overall, the paper connects intrinsic code geometry to classical group structures and classifies broad families of codes under group-code actions.

Abstract

A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism from FG to Fn which maps G to the standard basis of Fn. Many classical linear codes have been shown to be group codes. In this paper we obtain a criterion to decide when a linear code is a group code in terms of its intrinsical properties in the ambient space Fn, which does not assume an a priori group algebra structure on Fn. As an application we provide a family of groups (including metacyclic groups) for which every two-sided group code is an abelian group code. It is well known that Reed-Solomon codes are cyclic and its parity check extensions are elementary abelian group codes. These two classes of codes are included in the class of Cauchy codes. Using our criterion we classify the Cauchy codes of some lengths which are left group codes and the possible group code structures on these codes.

An intrinsical description of group codes

TL;DR

This work gives an intrinsic criterion to determine when a linear code is a group code, by analyzing the permutation automorphism group PAut(C) and locating regular subgroups within it. It shows that a code is a left G-code precisely when G corresponds to a regular, transitive subgroup of S_n contained in PAut(C), and a G-code when H also satisfies H ∪ C_{S_n}(H) ⊆ PAut(C). The authors apply this framework to 1-dimensional and Cauchy codes, deriving concrete structural results: (i) one-dimensional left G-codes require a block-structure with a normal subgroup leading to a cyclic quotient, (ii) any G-code with G = AB for abelian A,B is abelian, yet left G-codes can be non-abelian, and (iii) for q-ary Cauchy codes of length q or q−1 the possible left G-code structures are tightly controlled (cyclic or dihedral) with precise congruence conditions; in particular, many large-location codes reduce to cyclic forms when q ≡ −1 (mod 4). Overall, the paper connects intrinsic code geometry to classical group structures and classifies broad families of codes under group-code actions.

Abstract

A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism from FG to Fn which maps G to the standard basis of Fn. Many classical linear codes have been shown to be group codes. In this paper we obtain a criterion to decide when a linear code is a group code in terms of its intrinsical properties in the ambient space Fn, which does not assume an a priori group algebra structure on Fn. As an application we provide a family of groups (including metacyclic groups) for which every two-sided group code is an abelian group code. It is well known that Reed-Solomon codes are cyclic and its parity check extensions are elementary abelian group codes. These two classes of codes are included in the class of Cauchy codes. Using our criterion we classify the Cauchy codes of some lengths which are left group codes and the possible group code structures on these codes.
Paper Structure (4 sections, 14 theorems, 24 equations)

This paper contains 4 sections, 14 theorems, 24 equations.

Table of Contents

  1. Characterizing group codes
  2. One-dimensional group codes
  3. Group codes versus abelian group codes
  4. Cauchy codes

Key Result

Lemma 1.1

Let $H$ be a regular subgroup of $S_n$ and fix an element $i_0\in {\mathbb N}_n$. Let $\psi:H\rightarrow {\mathbb N}_n$ be the bijection given by $\psi(h)=h(i_0)$. Then there is an anti-isomorphism $\sigma:H\rightarrow C_{S_n}(H)$, mapping $h\in H$ to $\sigma_h$, where Moreover $\sigma_h=h$ for every $h\in Z(H)$ and so $Z(H)=Z(C_{S_n}(H))$.

Theorems & Definitions (28)

  • Lemma 1.1
  • proof
  • Theorem 1.2
  • proof
  • Corollary 1.3
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Theorem 3.1
  • ...and 18 more