合并机制下的可转换多项式评估码：斜多项式框架
Convertible Polynomial Evaluation Codes in the Merge Regime: A Skew-Polynomial Framework
AI总结：
本文开发合并机制下可转换码的斜多项式代数框架，提出两种斜多项式评估码转换模板，实现线性化里德-所罗门码、加比杜林码的合并转换，达到按符号访问最优成本。
AI中文摘要：
本文针对合并机制下的可转换码开发了一种代数框架，该框架直接作用于底层码的多项式评估结构。我们刻画了与共轭类的并集相关联的最小斜多项式，建立了与评估兼容的乘积法则，并推导了针对这些评估集的特殊中国剩余定理（sCRT）。基于这套机制，我们提出了两种用于斜多项式评估码（PECs）的转换模板：第一种适用于不同的初始PEC，其不同的评估集自然提供了sCRT所需的模；第二种处理相同的初始PEC，为此引入了不同的辅助评估集和显式代数相容性条件，以保持符号不变，并从指定的读取符号生成写入符号。当以标准基$1,x,\ldots,x^{k-1}$实例化时，两种构造均产生线性化里德-所罗门（Reed–Solomon）码的合并转换，其最终码等价于线性化里德-所罗门码，且实现了按符号访问最优成本。该框架进一步专门化到交换环$\mathbb{F}_q[x]$，从而得到普通PEC的对应转换构造，并将已知的里德-所罗门码和塔莫-巴尔（Tamo–Barg）码的多项式形式构造作为特例恢复。作为进一步的应用，据我们所知，这种专门化提供了第一种具有加比杜林（Gabidulin）初始码的合并机制可转换构造，其最终码等价于加比杜林码，且在本文考虑的符号访问模型下达到按符号访问最优性。
英文摘要：
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.
