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Information sets from defining sets for Reed-Muller codes of first and second order

José Joaquín Bernal, Juan Jacobo Simón

TL;DR

This work addresses the problem of constructing information sets for Reed-Muller codes of first and second order by leveraging their realization as affine-invariant abelian codes. The authors employ the defining-set perspective and apply the two-dimensional abelian-code method BS to obtain information sets directly from defining sets, translating cyclic-code information to punctured cyclic/abelian formats. They derive explicit, parameter-driven descriptions of information sets for $R(1,m)$ and $R(2,m)$ by presenting concrete formulas for the check-position sets and their preimages under group isomorphisms, with multiple options arising from different isomorphisms. The results facilitate potential permutation-decoding approaches and offer a bridge between algebraic structure and practical decoding, while highlighting the role of $2$-orbit structures and suitable representatives in the construction. Overall, the paper provides a modular, algebraically grounded pathway to information-set construction for RM codes, with clear generalizations and applications in decoding strategies.

Abstract

Reed-Muller codes belong to the family of affine-invariant codes. As such codes they have a defining set that determines them uniquely, and they are extensions of cyclic group codes. In this paper we identify those cyclic codes with multidimensional abelian codes and we use the techniques introduced in \cite{BS} to construct information sets for them from their defining set. For first and second order Reed-Muller codes, we describe a direct method to construct information sets in terms of their basic parameters.

Information sets from defining sets for Reed-Muller codes of first and second order

TL;DR

This work addresses the problem of constructing information sets for Reed-Muller codes of first and second order by leveraging their realization as affine-invariant abelian codes. The authors employ the defining-set perspective and apply the two-dimensional abelian-code method BS to obtain information sets directly from defining sets, translating cyclic-code information to punctured cyclic/abelian formats. They derive explicit, parameter-driven descriptions of information sets for and by presenting concrete formulas for the check-position sets and their preimages under group isomorphisms, with multiple options arising from different isomorphisms. The results facilitate potential permutation-decoding approaches and offer a bridge between algebraic structure and practical decoding, while highlighting the role of -orbit structures and suitable representatives in the construction. Overall, the paper provides a modular, algebraically grounded pathway to information-set construction for RM codes, with clear generalizations and applications in decoding strategies.

Abstract

Reed-Muller codes belong to the family of affine-invariant codes. As such codes they have a defining set that determines them uniquely, and they are extensions of cyclic group codes. In this paper we identify those cyclic codes with multidimensional abelian codes and we use the techniques introduced in \cite{BS} to construct information sets for them from their defining set. For first and second order Reed-Muller codes, we describe a direct method to construct information sets in terms of their basic parameters.
Paper Structure (8 sections, 19 theorems, 71 equations, 6 tables)

This paper contains 8 sections, 19 theorems, 71 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Information sets for abelian codes
  4. Cyclic codes as two-dimensional cyclic codes
  5. Reed-Muller codes
  6. Information sets for first-order Reed-Muller codes
  7. Information sets for second-order Reed-Muller codes
  8. Conclusions

Key Result

Theorem 9

Let $r_1, r_2$ be odd integers and let ${\mathcal{C}}$ be an abelian code in $\mathbb A(r_1,r_2)$ with defining set ${\mathcal{D}}_\alpha({\mathcal{C}})$ with respect to $\alpha=\{\alpha_1,\alpha_2\}$. Then $\Gamma({\mathcal{C}})$ is a set of check positions for ${\mathcal{C}}$.

Theorems & Definitions (58)

  • Definition 1
  • Remark 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Remark 7
  • Example 8
  • Theorem 9
  • Example 10
  • ...and 48 more