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Channel Coding based on Skew Polynomials and Multivariate Polynomials

Hedongliang Liu

TL;DR

This work introduces a unified algebraic framework for channel and storage coding that leverages skew polynomials and multivariate polynomials to craft dual-containing and repair-friendly codes. It develops both theoretical foundations and computational tools: (i) a general skew-polynomial apparatus for constructing dual-containing polycyclic codes over rings, (ii) a GM-MSRD condition and a distributed LRS scheme that achieves maximum sum-rank distance under support constraints, and (iii) an extensive exploration of multivariate evaluation codes (including QLRS) with local recovery and availability properties, plus joint decoding bounds for interleaved alternant codes. The results enable quantum-error-correcting compatible codes, efficient distributed storage, and improved vector-network-coding strategies with practical field-size benefits. Collectively, the work bridges non-commutative polynomial theory with concrete coding solutions for networks and quantum information, providing both constructive methods and decoding insights. The findings highlight how non-traditional polynomial structures open new avenues for high-distance, repair-enabled codes in realistic, large-scale systems.

Abstract

This dissertation considers new constructions and decoding approaches for error-correcting codes based on non-conventional polynomials, with the objective of providing new coding solutions to the applications mentioned above. With skew polynomials, we construct codes that are dual-containing, which is a desired property of quantum error-correcting codes. By considering evaluation codes based on skew polynomials, a condition on the existence of optimal support-constrained codes is derived and an application of such codes in the distributed multi-source networks is proposed. For a class of multicast networks, the advantage of vector network coding compared to scalar network coding is investigated. Multivariate polynomials have been attracting increasing interest in constructing codes with repair capabilities by accessing only a small amount of available symbols, which is required to build failure-resistant distributed storage systems. A new class of bivariate evaluation codes and their local recovery capability are studied. Interestingly, the well-known Reed-Solomon codes are used in a class of locally recoverable codes with availability (multiple disjoint recovery sets) via subspace design. Aside from new constructions, decoding approaches are considered in order to increase the error correction capability in the case where the code is fixed. In particular, new lower and upper bounds on the success probability of joint decoding interleaved alternant codes by a syndrome-based decoder are derived, where alternant codes are an important class of algebraic codes containing Goppa codes, BCH codes, and Reed-Muller codes as sub-classes.

Channel Coding based on Skew Polynomials and Multivariate Polynomials

TL;DR

This work introduces a unified algebraic framework for channel and storage coding that leverages skew polynomials and multivariate polynomials to craft dual-containing and repair-friendly codes. It develops both theoretical foundations and computational tools: (i) a general skew-polynomial apparatus for constructing dual-containing polycyclic codes over rings, (ii) a GM-MSRD condition and a distributed LRS scheme that achieves maximum sum-rank distance under support constraints, and (iii) an extensive exploration of multivariate evaluation codes (including QLRS) with local recovery and availability properties, plus joint decoding bounds for interleaved alternant codes. The results enable quantum-error-correcting compatible codes, efficient distributed storage, and improved vector-network-coding strategies with practical field-size benefits. Collectively, the work bridges non-commutative polynomial theory with concrete coding solutions for networks and quantum information, providing both constructive methods and decoding insights. The findings highlight how non-traditional polynomial structures open new avenues for high-distance, repair-enabled codes in realistic, large-scale systems.

Abstract

This dissertation considers new constructions and decoding approaches for error-correcting codes based on non-conventional polynomials, with the objective of providing new coding solutions to the applications mentioned above. With skew polynomials, we construct codes that are dual-containing, which is a desired property of quantum error-correcting codes. By considering evaluation codes based on skew polynomials, a condition on the existence of optimal support-constrained codes is derived and an application of such codes in the distributed multi-source networks is proposed. For a class of multicast networks, the advantage of vector network coding compared to scalar network coding is investigated. Multivariate polynomials have been attracting increasing interest in constructing codes with repair capabilities by accessing only a small amount of available symbols, which is required to build failure-resistant distributed storage systems. A new class of bivariate evaluation codes and their local recovery capability are studied. Interestingly, the well-known Reed-Solomon codes are used in a class of locally recoverable codes with availability (multiple disjoint recovery sets) via subspace design. Aside from new constructions, decoding approaches are considered in order to increase the error correction capability in the case where the code is fixed. In particular, new lower and upper bounds on the success probability of joint decoding interleaved alternant codes by a syndrome-based decoder are derived, where alternant codes are an important class of algebraic codes containing Goppa codes, BCH codes, and Reed-Muller codes as sub-classes.
Paper Structure (105 sections, 110 theorems, 418 equations, 13 figures, 12 tables, 8 algorithms)

This paper contains 105 sections, 110 theorems, 418 equations, 13 figures, 12 tables, 8 algorithms.

Table of Contents

  1. Motivation and Overview
  2. Introduction to Codes based on Polynomials
  3. Basic Notations
  4. Multivariate Polynomials
  5. Ideals and Variety
  6. Gröbner Bases
  7. Skew Polynomials
  8. The General Definition: $\mathcal{A}[X;\theta,\delta]$
  9. Multiplication
  10. Division
  11. Evaluation
  12. Ideals
  13. With Frobenius Automorphism and Zero Derivation
  14. $\sigma$-Conjugacy Classes
  15. Polynomial Independence
  16. ...and 90 more sections

Key Result

Theorem 2.1

Let $F$ be an arbitrary field and $f$ be a nonzero polynomial in $F[x_1,\dots,x_n]$ of total degree $\deg(f)=\sum_{i=1}^n t_i$, where $t_i\in\mathbb{N}, \ \forall i$. Then, if $\mathcal{X}_1,\dots, \mathcal{X}_n$ are subsets of $F$ with $|\mathcal{X}_i|> t_i$, then there are $\hat{x}_1\in \mathcal{X

Figures (13)

  • Figure 1: Transmission model via a linear $[n,k]$ block code with an alphabet $\mathcal{A}$.
  • Figure 2: Proof logic for $\ref{['item:equiv2']}\implies\ref{['item:equiv1']}$ with initial hypothesis \ref{['H:fromPre']} and \ref{['H:fromHere']}.
  • Figure 3: Illustration of the induction for \ref{['item:equiv2']}$\implies$\ref{['item:equiv1']} under difference cases.
  • Figure 4: Illustration of the distributed multi-source network model.
  • Figure 5: Illustration of the required encoding matrix for the instance of distributed multi-source networks with $h=4$ messages. This support-constrained matrix is a generator matrix of a $[23,9]$ LRS over $\mathbb{F}_{4^9}$ with $\ell=3$ blocks.
  • ...and 8 more figures

Theorems & Definitions (257)

  • Definition 2.1: Monomial order
  • Definition 2.2
  • Theorem 2.1: Combinatorial Nullstellensatz alon1999combinatorial
  • Definition 2.3: Gröbner basis
  • Theorem 2.2: von2013modern
  • Definition 2.4: $S$-polynomial
  • Theorem 2.3: von2013modern
  • Definition 2.5
  • Definition 2.6: Endomorphism and derivation
  • Lemma 2.1
  • ...and 247 more