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The Dimension of Subcode-Subfields of Shortened Generalized Reed Solomon Codes

Fernando Hernando, Kyle Marshall, Michael E. O'Sullivan

TL;DR

The paper addresses constructing high-dimension subfield-subcodes of shortened Generalized Reed-Solomon codes by leveraging the trace map and cyclotomic twist polynomials. It develops a theoretical framework that expresses SFSC dimensions via the kernel of a trace-derived map on a dual GRS code, and derives a concrete lower bound using cyclotomic coset decomposition. Two search algorithms—one based on cyclotomic twists and another on puncturing patterns—are used to enumerate new codes, yielding 98 codes with improved parameters across $\mathbb{F}_2$, $\mathbb{F}_3$, and $\mathbb{F}_5$, including several direct new codes and many shortened/punctured variants. This work broadens the catalog of good subfield-subcodes and provides practical methods to tailor RS-derived codes with larger dimensions for fixed lengths. The approach has potential impact on applications requiring fixed-length, high-dimension SFSCs with reliable minimum distance properties.

Abstract

Reed-Solomon (RS) codes are among the most ubiquitous codes due to their good parameters as well as efficient encoding and decoding procedures. However, RS codes suffer from having a fixed length. In many applications where the length is static, the appropriate length can be obtained by an RS code by shortening or puncturing. Generalized Reed-Solomon (GRS) codes are a generalization of RS codes, whose subfield-subcodes are extensively studied. In this paper we show that a particular class of GRS codes produces many subfield-subcodes with large dimension. An algorithm for searching through the codes is presented as well as a list of new codes obtained from this method.

The Dimension of Subcode-Subfields of Shortened Generalized Reed Solomon Codes

TL;DR

The paper addresses constructing high-dimension subfield-subcodes of shortened Generalized Reed-Solomon codes by leveraging the trace map and cyclotomic twist polynomials. It develops a theoretical framework that expresses SFSC dimensions via the kernel of a trace-derived map on a dual GRS code, and derives a concrete lower bound using cyclotomic coset decomposition. Two search algorithms—one based on cyclotomic twists and another on puncturing patterns—are used to enumerate new codes, yielding 98 codes with improved parameters across , , and , including several direct new codes and many shortened/punctured variants. This work broadens the catalog of good subfield-subcodes and provides practical methods to tailor RS-derived codes with larger dimensions for fixed lengths. The approach has potential impact on applications requiring fixed-length, high-dimension SFSCs with reliable minimum distance properties.

Abstract

Reed-Solomon (RS) codes are among the most ubiquitous codes due to their good parameters as well as efficient encoding and decoding procedures. However, RS codes suffer from having a fixed length. In many applications where the length is static, the appropriate length can be obtained by an RS code by shortening or puncturing. Generalized Reed-Solomon (GRS) codes are a generalization of RS codes, whose subfield-subcodes are extensively studied. In this paper we show that a particular class of GRS codes produces many subfield-subcodes with large dimension. An algorithm for searching through the codes is presented as well as a list of new codes obtained from this method.

Paper Structure

This paper contains 8 sections, 13 theorems, 27 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Construction of Punctured Reed-Solomon Codes
  4. The Dimension of Subfield-Subcodes of Certain Shortened GRS Codes
  5. Computation
  6. New codes over $\mathbb{F}_2$
  7. New codes over $\mathbb{F}_3$
  8. New codes over $\mathbb{F}_5$

Key Result

Lemma 2.3

Let $R = \mathbb{F}_{p^m}[x]/\langle x^N - 1\rangle$ and $\alpha = (1,\eta,...,\eta^{N-1})$ where $\eta$ is a primitive element in $\mathbb{F}_{p^m}$. Then, $R$ is isomorphic to $\mathbb{F}_{p^m}^N$ under the map $\textnormal{Ev}$, where $\textnormal{Ev}^{-1}$is given by Lagrange Interpolation.

Figures (6)

  • Figure 1: New Codes over $\mathbb{F}_2$ obtained with Algorithm \ref{['alg1']}
  • Figure 2: New Codes over $\mathbb{F}_2$ obtained from $C_1$ and $C_2$
  • Figure 3: New Codes over $\mathbb{F}_3$ obtained with Algorithm \ref{['alg1']}
  • Figure 4: New Codes over $\mathbb{F}_3$ obtained $C_{20},\ldots,C_{33}$
  • Figure 5: New Codes over $\mathbb{F}_5$ obtained with Algorithm \ref{['alg1']}
  • ...and 1 more figures

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Definition 2.5
  • Theorem 2.6: Delsarte
  • Definition 2.7
  • Lemma 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 16 more