Convertible Polynomial Evaluation Codes in the Merge Regime: A Skew-Polynomial Framework

2026-08-25

AI Summary

本文提出了一种基于斜多项式的框架，用于解决合并机制下的可转换代码问题，并通过特定的中国剩余定理和两种转换模板实现线性化Reed-Solomon码及Gabidulin码之间的高效转换。

Abstract

In this paper, we develop an algebraic framework for merge-regime convertible codes that works directly with the polynomial-evaluation structure of the underlying codes. We characterize the minimal skew polynomials associated with unions of conjugacy classes, establish an evaluation-compatible product rule, and derive a special Chinese Remainder Theorem (sCRT) tailored to these evaluation sets. Based on this machinery, we propose two conversion templates for skew polynomial evaluation codes (PECs). The first applies to distinct initial PECs, whose different evaluation sets naturally provide the moduli required by sCRT. The second treats identical initial PECs, for which distinct auxiliary evaluation sets and explicit algebraic compatibility conditions are introduced to preserve unchanged symbols and to generate written symbols from designated read symbols. When instantiated with the standard basis $1,x,\ldots,x^{k-1}$, both constructions yield merge conversions for linearized Reed--Solomon codes whose final codes are equivalent to linearized Reed--Solomon codes and achieve per-symbol access-optimal cost. The framework further specializes to the commutative ring $\mathbb{F}_q[x]$, yielding corresponding conversion constructions for ordinary PECs and recovering the known polynomial-form constructions for Reed--Solomon and Tamo--Barg codes as special cases. As a further application, to the best of our knowledge, this specialization gives the first merge-regime convertible construction with Gabidulin initial codes and a final code equivalent to a Gabidulin code, while attaining per-symbol optimal access under the symbol-access model considered in this paper.

Problem

• Convertible Codes
• Skew Polynomial
• Evaluation Codes
• Chinese Remainder Theorem
• Reed-Solomon Codes

Innovation

• skew polynomial
• convertible codes
• evaluation-compatible product rule
• special Chinese Remainder Theorem (sCRT)
• Gabidulin codes