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Zero-divisor codes are codes constructed using group rings where their generators are zero-divisors. Generally, zero-divisor codes can be equivalent despite their associated groups are non-isomorphic, leading to the proposed conjecture "Every dihedral zero-divisor code has an equivalent form of cyclic zero-divisor code". This paper is devoted to study equivalence of zero-divisor codes in F 2 G having generators from the 2-nilradical of F 2 G , consisting of all nilpotents of nilpotency degree 2 of F 2 G . Essentially, algebraic structures of 2-nilradicals are first studied in general for both commutative and non-commutative F 2 G before specialized into the case when G is cyclic and dihedral. Then, results are used to study the conjecture above in the cases where the codes generators are from their respective 2-nilradicals. [ABSTRACT FROM AUTHOR]

In this article, we present a new construction of evaluation codes in the Hamming metric, which we call twisted Reed–Solomon codes. Whereas Reed–Solomon (RS) codes are MDS codes, this need not be the case for twisted RS codes. Nonetheless, we show that our construction yields several families of MDS codes. Further, for a large subclass of (MDS) twisted RS codes, we show that the new codes are not generalized RS codes. To achieve this, we use properties of Schur squares of codes as well as an explicit description of the dual of a large subclass of our codes. We conclude the paper with a description of a decoder, that performs very well in practice as shown by extensive simulation results. [ABSTRACT FROM AUTHOR]

A q -ary code of length n , size M , and minimum distance d code. An (n,q^{k},n-k+1)_{q} code is called a maximum distance separable (MDS) code. In this paper, some MDS codes over small alphabets are classified. It is shown that every (k+d-1,q^{k},d)_{q} code with k\geq 3 , d \geq 3 , is equivalent to a linear code with the same parameters. This implies that the (6,5^4,3)5 code and the (n,7^{n-2},3)7 MDS codes for n\in \6,7,8\ are unique. The classification of one-error-correcting 8-ary MDS codes is also finished; there are 14, 8, 4, and 4 equivalence classes of (n,8^n-2,3)8 codes for $n=6,7,8$ , and 9, respectively. One of the equivalence classes of perfect (9,8^{7},3)_{8} codes corresponds to the Hamming code and the other three are nonlinear codes for which there exists no previously known construction. [ABSTRACT FROM PUBLISHER]

Starting from the problem of d-tensor isomorphism (d- TI ), we study the relation between various code equivalence problems in different metrics. In particular, we show a reduction from the sum-rank metric ( CE sr ) to the rank metric ( CE rk ). To obtain this result, we investigate reductions between tensor problems. We define the monomial isomorphism problem for d-tensors (d- TI ∗ ), where, given two d-tensors, we ask if there are d - 1 invertible matrices and a monomial matrix sending one tensor into the other. We link this problem to the well-studied d- TI and the TI -completeness of d- TI ∗ is shown. Due to this result, we obtain a reduction from CE sr to CE rk . In the literature, a similar result was known, but it needs an additional assumption on the automorphisms of matrix codes. Since many constructions based on the hardness of Code Equivalence problems are emerging in cryptography, we analyze how such reductions can be taken into account in the design of cryptosystems based on CE sr . [ABSTRACT FROM AUTHOR]

An algorithm for equivalence of linear codes over finite fields is presented. Its main advantage is that it can extract exactly one representative from each equivalence class among a large number of linear codes. It can also be used as a test for isomorphism of binary matrices. The algorithm is implemented in the program LCequivalence, which is designed to obtain the inequivalent codes in a set of linear codes over a finite field with q < 64 elements. This program is a module of the free software package QextNewEdition for constructing, classifying and studying linear codes. [ABSTRACT FROM AUTHOR]

A pair (C, D) of group codes in R[G] is called a linear complementary pair (abbreviated to LCP) if C ⊕ D = R [ G ] , where R is a finite Frobenius ring, and G is a finite group. We provide a necessary and sufficient condition for a pair (C, D) of group codes in R[G] to be LCP. Furthermore, we prove that if (C, D) is an LCP of group codes in R[G], then C and D ⊥ are permutation equivalent. [ABSTRACT FROM AUTHOR]

Two algorithms for the classification of linear codes over finite fields are presented. One of the algorithms is based on canonical augmentation and the other one on lattice point enumeration. New classification results over fields with 2, 3 and 4 elements are obtained. [ABSTRACT FROM AUTHOR]