compute weight enumerator

This work investigates the weight distribution of cascaded code ensembles based on the Plotkin construction, aiming to establish an exact mathematical relationship between the ensemble’s weight distribution and that of its constituent component codes. For Reed–Muller (RM)-like structured codes, we propose the first unified theoretical framework that explicitly expresses the overall weight distribution as a combinatorial function of the component codes’ weight distributions. Methodologically, we integrate generating functions, combinatorial enumeration, and algebraic coding analysis, and introduce a probabilistic model to characterize the statistical behavior of codeword weights under the cascade architecture. Our key contribution is a closed-form, efficiently computable expression for the weight distribution, enabling asymptotic performance evaluation and decoding error probability analysis for RM-like codes. This result significantly deepens the theoretical foundations of classical Plotkin-type constructions and broadens their practical applicability.

本文通过单体多项式形式化方法，介绍极化码的权重结构基础，系统地描述并列举低权重码字，为读者深入理解极化码权重分布做准备。

This study addresses the efficient computation of the minimum weight and weight enumerator for error-correcting codes generated by integer matrices. By interpreting the weight enumerator as a quasi-polynomial, the work establishes—for the first time—a transformational relationship with the Tutte quasi-polynomial. Leveraging reductions modulo integer residue rings and the characteristic quasi-polynomial of hyperplane arrangements, the counting of maximum-weight codewords is recast as a problem in combinatorial geometry. This approach substantially reduces the computational complexity of determining minimum weights and yields exact counts of maximum-weight codewords for codes associated with the $N_k$ and $Z_k$ matroids. The results provide a novel algebraic–geometric perspective within combinatorial coding theory.

This work addresses coding problems over finite fields, finite abelian groups, and finite Frobenius rings by systematically constructing generalized weight enumerators based on arbitrary equivalence relations. Methodologically, it integrates algebraic coding theory, group representation theory, and combinatorics to characterize the necessary and sufficient conditions under which such enumerators satisfy MacWilliams identities—establishing, for the first time, an equivalence-relation-driven framework for weight counting. By moving beyond classical reliance on Hamming or Lee metrics, the approach develops a unified duality analysis paradigm applicable across diverse algebraic structures. The theory’s completeness and consistency are rigorously verified under several canonical equivalence relations, including coordinate permutations and group-action orbits. The results substantially extend the scope of MacWilliams theory, providing novel analytical and constructive tools for non-standard code spaces.

This study investigates the failure of the MacWilliams identity under non-Hamming weight functions, demonstrating that distinct linear codes can share identical weight enumerators while their duals exhibit different weight enumerators. By integrating algebraic coding theory, finite field analysis, and comparative studies of weight enumerator functions, this work provides the first systematic characterization of weight functions for which the MacWilliams identity does not hold, thereby extending beyond the classical Hamming-weight framework. The authors construct multiple families of counterexamples, revealing the widespread existence of such pathological weights and significantly deepening the understanding of duality properties of linear codes under generalized weight functions.

This study investigates the weight distribution of the third-order Reed–Muller code RM(3,11) of length 2048 and the covering radii of certain subcodes. By analyzing the 3,691,560 nonzero orbit representatives of ternary Boolean cubic forms under the action of GL(10,2), and leveraging coset weight enumerators together with structural theorems—specifically, that every nondegenerate Boolean cubic form admits a nondegenerate hyperplane restriction except for a unique orbit in odd dimensions—the complete weight distribution of RM(3,11) is determined. Furthermore, the relative covering radius of RM(2,10) in RM(3,10) is precisely improved to 408, and the upper bound on the relative covering radius of RM(6,10) in RM(7,10) is significantly reduced from 50 to 32.

本文研究了二进制单奇偶校验积码的结构和计数特性，通过特征和方法确定了其权重分布，解决了无需枚举所有码字即可计算全权重分布的问题。

This study investigates the weight distribution of generalized doubly extended Reed–Solomon codes with minimum distance \(d \geq 5\) over cosets of weight 2. By leveraging finite field theory, orbit analysis under group actions, and algebraic combinatorial techniques, the authors establish for the first time that Case S holds—implying the code is 2-regular—whenever \(q-1\) and \(d-2\) are coprime. They further introduce two equivalent new combinatorial problems, denoted \(A_{q,\mu}^\times\) and \(A_{R,\mu}^+\), which provide a viable pathway toward resolving Case NS. The proposed general framework successfully computes complete weight distributions for multiple \((q,d)\) parameter pairs, thereby solving the long-standing problem of determining sufficient conditions for Case S.

This study addresses the long-standing open problem of determining the weight distribution of Reed–Muller codes $RM(m-7,m)$, with a focus on the pivotal case $RM(7,14)$. By integrating algebraic coding theory, combinatorial analysis, and efficient computational enumeration techniques, the authors nearly completely resolve the weight spectrum of $RM(7,14)$, leaving only eight weights undetermined. This achievement substantially advances the understanding of the structural properties of high-order Reed–Muller codes and provides crucial theoretical foundations for characterizing the weight distribution in the general family $RM(m-7,m)$. The work thus represents a major breakthrough in the case $c=7$, significantly narrowing the gap toward a full resolution of this classical problem in coding theory.

该研究通过归纳法和特定工具计算，完全确定了几类广义Reed-Muller码的重量谱。