摘要:
该文研究了有限域$ {\mathbb{F}}_{q} $的一类双阶扭曲广义里德-所罗门(GRS)码$ {C}_{k,\boldsymbol{h},\boldsymbol{\eta }}(\boldsymbol{\alpha },\boldsymbol{v}) $及其扩展码$ {C}_{k,\boldsymbol{h},\boldsymbol{\eta }}(\boldsymbol{\alpha },\boldsymbol{v},\mathrm{\infty }) $，不仅给出了这两类码的校验矩阵，还分别刻画了码$ {C}_{k,\boldsymbol{h},\boldsymbol{\eta }}(\boldsymbol{\alpha },\boldsymbol{v}) $是极大距离可分(MDS)码或者是几乎极大距离可分(AMDS)码以及码$ {C}_{k,\boldsymbol{h},\boldsymbol{\eta }}(\boldsymbol{\alpha },\boldsymbol{v},\mathrm{\infty }) $是MDS码的充要条件。基于舒尔方法，当$ k\geq 4 $时，该文确定了这两类码的非GRS性质，还分别给出了码$ {C}_{k,\boldsymbol{h},\boldsymbol{\eta }}(\boldsymbol{\alpha },\boldsymbol{v}) $为几乎自对偶码以及码$ {C}_{k,\boldsymbol{h},\boldsymbol{\eta }}(\boldsymbol{\alpha },\boldsymbol{v},\mathrm{\infty }) $为自正交码的充要条件，并且构造了一类具有灵活参数的几乎自对偶双阶扭曲GRS码。
Abstract:
Objective In the field of coding theory, Twisted Generalized Reed-Solomon (TGRS) codes have attracted considerable research interest for their flexible structural properties. However, investigations into their extended codes remain relatively limited. Existing literature indicates that prior studies on extended TGRS codes are scarce, with only a few works delving into this area, thereby leaving significant gaps in our understanding of their error-correcting capabilities, duality properties, and practical applications. Meanwhile, the foundational parity-check matrix forms for TGRS codes presented in earlier research lack sufficient clarity and exhibit restricted parameter coverage. Specifically, previous studies fail to accommodate scenarios involving h=0, which constrains their utility in broader coding scenarios where diverse parameter configurations are required. Furthermore, constructing non-GRS codes is an intriguing and critical research topic due to their unique characteristics to resist Sidelnikov-Shestakov and Wieschebrink attacks, whereas GRS codes are vulnerable to such threats. Additionally, Maximum Distance Separable (MDS) codes, self-orthogonal codes, and almost self-dual codes are highly valued for their efficient error-correcting capabilities and structural advantages. MDS codes, achieving the Singleton bound, are essential for distributed storage systems where data integrity under node failures is critical; self-orthogonal and almost self-dual codes, with their inherent duality properties, play key roles in quantum coding, secret sharing schemes, and secure multi-party computation, where structural regularity and cryptographic security are vital. Accordingly, this paper aims to achieve the following goals: (1) characterize the MDS and Almost MDS (AMDS) properties of double-twisted GRS codes$ {C}_{k,\boldsymbol{h},\boldsymbol{\eta }}(\boldsymbol{\alpha },\boldsymbol{v}) $and their extended codes$ {C}_{k,\boldsymbol{h},\boldsymbol{\eta }}(\boldsymbol{\alpha },\boldsymbol{v},\mathrm{\infty }) $; (2) derive explicit and unified parity-check matrices applicable to all valid parameter ranges, including h=0; (3) establish non-GRS properties of these codes under specific parameter conditions; (4) provide rigorous necessary and sufficient conditions for the extended codes to be self-orthogonal and for the original codes to be almost self-dual; and (5) construct a class of almost self-dual double-twisted GRS codes with flexible parameters to meet diverse application requirements in secure and reliable communication systems. Methods The research adopts a comprehensive framework rooted in algebraic coding theory and finite field mathematics. Algebraic Analysis serves as a foundational tool: explicit parity-check matrices are derived using properties of polynomial rings over finite fields $ {F}_{q} $, Vandermonde matrices structures, and polynomial interpolation techniques; The Schur Product Method is utilized to determine non-GRS properties by evaluating the dimension of the Schur square of codes and their duals, distinguishing them from GRS codes; Linear Algebra and Combinatorics are utilized to characterize MDS and AMDS properties. By examining the non-singularity of generator matrix submatrices and solving systems of equations involving symmetric sums of finite field elements, the conditions for MDS and AMDS codes are derived. These conditions rely on sets$ {S}_{k}(\boldsymbol{\alpha },\boldsymbol{\eta }) $,$ {L}_{k}(\boldsymbol{\alpha },\boldsymbol{\eta }) $, and$ {D}_{k}(\boldsymbol{\alpha },\boldsymbol{\eta }) $, which are defined based on sums of products of finite field elements. Duality theory forms the foundation for analyzing orthogonality. For self-orthogonal codes$ C\subseteq {C}^{\bot } $, the generator matrix must satisfy$ G{G}^{\rm T}=\boldsymbol{O} $. For almost self-dual codes (length-odd, dimension-(n-1)/2 self-orthogonal codes), this condition is combined with structural properties of dual codes and symmetric sum relations of $ {\alpha }_{i} $ to derive necessary and sufficient conditions. Results and Discussions For MDS and AMDS properties, critical findings are established: The extended double-twisted GRS code$ {C}_{k,\boldsymbol{h},\boldsymbol{\eta }}(\boldsymbol{\alpha },\boldsymbol{v},\mathrm{\infty }) $is MDS if and only if$ 1\notin {S}_{k}(\boldsymbol{\alpha },\boldsymbol{\eta }) $and$ 1\notin {L}_{k}(\boldsymbol{\alpha },\boldsymbol{\eta }) $; the double-twisted GRS code$ {C}_{k,\boldsymbol{h},\boldsymbol{\eta }}(\boldsymbol{\alpha },\boldsymbol{v}) $is AMDS if and only if$ 1\in {S}_{k}(\boldsymbol{\alpha },\boldsymbol{\eta }) $and$ (0,1)\notin {D}_{k}(\boldsymbol{\alpha },\boldsymbol{\eta }) $; and$ {C}_{k,\boldsymbol{h},\boldsymbol{\eta }}(\boldsymbol{\alpha },\boldsymbol{v}) $is neither MDS nor AMDS if and only if$ (0,1)\in {D}_{k}(\boldsymbol{\alpha },\boldsymbol{\eta }) $. Unified parity-check matrices of$ {C}_{k,\boldsymbol{h},\boldsymbol{\eta }}(\boldsymbol{\alpha },\boldsymbol{v}) $and$ {C}_{k,\boldsymbol{h},\boldsymbol{\eta }}(\boldsymbol{\alpha },\boldsymbol{v},\mathrm{\infty }) $ for all$ 0\leq h\leq k-1 $are derived, resolving prior limitations that excluded h=0 by removing restrictive submatrix structure constraints. For non-GRS properties, when$ k\geq 4 $and$ n-k\geq 4 $, $ {C}_{k,\boldsymbol{h},\boldsymbol{\eta }}(\boldsymbol{\alpha },\boldsymbol{v}) $and its extened codes $ {C}_{k,\boldsymbol{h},\boldsymbol{\eta }}(\boldsymbol{\alpha },\boldsymbol{v},\mathrm{\infty }) $are non-GRS regardless of$ 2k\geq n $or$ 2k \lt n $, confirmed by the dimension of their Schur squares exceeding that of corresponding GRS codes. This ensures resistance to Sidelnikov-Shestakov and Wieschebrink attacks. Regarding self-orthogonality and almost self-duality, the extended code$ {C}_{k,\boldsymbol{h},\boldsymbol{\eta }}(\boldsymbol{\alpha },\boldsymbol{v},\mathrm{\infty }) $with$ h=k-1 $is self-orthogonal under specific algebraic conditions;$ {C}_{k,\boldsymbol{h},\boldsymbol{\eta }}(\boldsymbol{\alpha },\boldsymbol{v}) $with$ h=k-1 $and$ n=2k+1 $is almost self-dual if and only if there exists$ \lambda \in F_{q}^{*} $such as$ \lambda {u}_{j}=v_{j}^{2} (j=1,\cdots ,2k+1) $and a symmetric sum constraint on$ {\alpha }_{i} $involving$ {\eta }_{1} $and$ {\eta }_{2} $holds. For odd prime power$ q $, a flexible almost self-dual code with parameters$ [q-t-1,(q-t-2)/2,\geq (q-t-2)/2] $is constructed using roots of $ m(x)=({x}^{q}-x)/f(x) $ where $ f(x)={x}^{t+1}-x $, with an example over$ {F}_{11} $yielding a$ [5,2,\geq 2] $code. Conclusions This work advances the study of double-twisted GRS codes and their extensions through key contributions: (1) complete characterization of MDS and AMDS properties via explicit combinatorial sets$ {S}_{k} $,$ {L}_{k} $,$ {D}_{k} $, enabling precise error-correcting capability assessment; (2) derivation of unified, explicit parity-check matrices for all$ 0\leq h\leq k-1 $, overcoming prior parameter restrictions and enhancing practical utility; (3) proof of non-GRS properties for$ k\geq 4 $, ensuring security against specific attacks; (4) rigorous conditions for self-orthogonal extended codes and almost self-dual original codes, deepening structural insights; (5) a flexible construction of almost self-dual codes, meeting diverse needs in secure communication and distributed storage. These results enrich coding theory and provide practical tools for robust, secure coding system design.
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