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Linearized Polynomial Chinese remainder codes

Philippe Gaborit, Camille Garnier, Olivier Ruatta

TL;DR

This work introduces linearized CRT ($q$-CRT) codes, a novel family of rank and sum-rank metric codes built from linearized polynomials and an effective Chinese Remainder Theorem. By embedding CRT lifting into the non-commutative ring of linearized polynomials, the authors construct a broad class of codes with explicit parity-check structures and connections to Gabidulin and skew Reed-Solomon codes. They propose a first decoding algorithm with a quantified failure probability for a special case, and extend the approach to a wider class of $q$-CRT codes, including probabilistic analysis of success and lifting behavior. The results offer a flexible parameter space, enabling tailored code characteristics for applications in distributed storage, cryptography, and communications, with concrete encoding/decoding procedures and performance tradeoffs highlighted.

Abstract

In this paper, we introduce a new family of codes relevent for rank and sum-rank metrics. These codes are based on an effective Chinese remainders theorem for linearized polynomials over finite fields. We propose a decoding algorithm for some instances of these codes.

Linearized Polynomial Chinese remainder codes

TL;DR

This work introduces linearized CRT (-CRT) codes, a novel family of rank and sum-rank metric codes built from linearized polynomials and an effective Chinese Remainder Theorem. By embedding CRT lifting into the non-commutative ring of linearized polynomials, the authors construct a broad class of codes with explicit parity-check structures and connections to Gabidulin and skew Reed-Solomon codes. They propose a first decoding algorithm with a quantified failure probability for a special case, and extend the approach to a wider class of -CRT codes, including probabilistic analysis of success and lifting behavior. The results offer a flexible parameter space, enabling tailored code characteristics for applications in distributed storage, cryptography, and communications, with concrete encoding/decoding procedures and performance tradeoffs highlighted.

Abstract

In this paper, we introduce a new family of codes relevent for rank and sum-rank metrics. These codes are based on an effective Chinese remainders theorem for linearized polynomials over finite fields. We propose a decoding algorithm for some instances of these codes.

Paper Structure

This paper contains 22 sections, 24 theorems, 34 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Rank and sum-rank metric codes
  3. Linearized polynomials
  4. Ring structure
  5. Link with endomorphism of $\mathbb{F}_{q^m}$
  6. Effective Chinese Remainder Theorem for linearized polynomials
  7. Linearized Polynomial Chinese Remainder Theorem codes
  8. Main properties of $q$CRT codes
  9. Some special constructions and examples
  10. Link with Gabidulin codes
  11. An instance with efficient encoding and efficient lifting
  12. Parity Check Matrices for $q$-CRT Codes
  13. Two-Blocks Case
  14. General Case
  15. A special case and a first decoding algorithm
  16. ...and 7 more sections

Key Result

Theorem 1

The map $d_r$ is a distance on $\operatorname{End} (V, W)$.

Figures (7)

  • Figure 1: We can observe that the probability of success is close to one for all rank weights that the algorithm can decode, i.e. all the rank weights smaller than the bound given by the linear system.
  • Figure 2: In this example, the bound given by the linear system is smaller than the unique decoding radius. The probability is close to one for all rank weights smaller than this bound.
  • Figure 3: In this example, the bound given by the linear system is larger than the unique decoding radius. The probability is close to one for all rank weights smaller than this bound.
  • Figure 4: In this example, the bound given by the linear system is larger than the unique decoding radius. However, the probability drops for rank weights below this bound, indicating that the algorithm may fail before reaching the limit imposed by the linear system.
  • Figure 5: For small values of $l$, the probability remains close to 1 for all rank weights smaller than the bound of the linear system.
  • ...and 2 more figures

Theorems & Definitions (34)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 1
  • Definition 6
  • Theorem 2
  • Lemma 2
  • ...and 24 more