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Determining hulls of generalized Reed-Solomon codes from algebraic geometry codes

Xue Jia, Qin Yue, Huan Sun, Junzhen Sui

TL;DR

This work advances understanding of hulls of generalized Reed-Solomon codes by modeling GRS codes and their duals as rational algebraic geometry codes and establishing when the hull is itself a GRS code through degree and gcd conditions on associated polynomials $s(z)$ and $t(z)$. When the geometric conditions are not met, it provides an explicit linear-algebra framework to compute hull bases and dimensions, ensuring practical computation. The study also furnishes constructions of self-orthogonal and self-dual GRS codes via these hull conditions, and it demonstrates the tightness of the proposed criteria with examples. Collectively, the results deepen the link between GRS codes and AG-code representations, with implications for LCD/self-dual code design and potential Hermitian hull analysis.

Abstract

In this paper, we provide conditions that hulls of generalized Reed-Solomon (GRS) codes are also GRS codes from algebraic geometry codes. If the conditions are not satisfied, we provide a method of linear algebra to find the bases of hulls of GRS codes and give formulas to compute their dimensions. Besides, we explain that the conditions are too good to be improved by some examples. Moreover, we show self-orthogonal and self-dual GRS codes.

Determining hulls of generalized Reed-Solomon codes from algebraic geometry codes

TL;DR

This work advances understanding of hulls of generalized Reed-Solomon codes by modeling GRS codes and their duals as rational algebraic geometry codes and establishing when the hull is itself a GRS code through degree and gcd conditions on associated polynomials and . When the geometric conditions are not met, it provides an explicit linear-algebra framework to compute hull bases and dimensions, ensuring practical computation. The study also furnishes constructions of self-orthogonal and self-dual GRS codes via these hull conditions, and it demonstrates the tightness of the proposed criteria with examples. Collectively, the results deepen the link between GRS codes and AG-code representations, with implications for LCD/self-dual code design and potential Hermitian hull analysis.

Abstract

In this paper, we provide conditions that hulls of generalized Reed-Solomon (GRS) codes are also GRS codes from algebraic geometry codes. If the conditions are not satisfied, we provide a method of linear algebra to find the bases of hulls of GRS codes and give formulas to compute their dimensions. Besides, we explain that the conditions are too good to be improved by some examples. Moreover, we show self-orthogonal and self-dual GRS codes.
Paper Structure (9 sections, 17 theorems, 123 equations)

This paper contains 9 sections, 17 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rational function field
  4. Rational function field and place
  5. Divisor, the Riemann-Roch space, the Riemann-Roch Theorem
  6. Rational algebraic geometry code
  7. Hulls of generalized Reed-Solomon Codes
  8. Self-orthogonal and self-dual GRS codes
  9. Conclusion and future work

Key Result

Theorem 2.1

(Riemann-Roch Theorem) (S) Let $W$ be a canonical divisor of $F/\Bbb F_q$. Then for each divisor $A\in Div(F)$,

Theorems & Definitions (41)

  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Remark 2.5
  • Lemma 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • Proposition 3.1
  • ...and 31 more