Related papers: On Z4-linear Reed-Muller like codes
Reed-Muller (RM) codes exhibit good performance under maximum-likelihood (ML) decoding due to their highly-symmetric structure. In this paper, we explore the question of whether the code symmetry of RM codes can also be exploited to achieve…
A cyclic codes of length $n$ over the rings $Z_{2^{m}}$ of integer of modulo $2^{m}$ is a linear code with property that if the codeword $(c_0,c_1,...,c_{n-1})\in \mathcal{C}$ then the cyclic shift $(c_1,c_2,...,c_0)\in \mathcal{C}$.…
Let $\mathbb{Z}_4$ denote the ring of integers modulo $4$. The Galois ring GR$(4,m)$, which consists of $4^m$ elements, represents the Galois extension of degree $m$ over $\mathbb{Z}_4$. The constructions of codes over $\mathbb{Z}_4$ have…
We investigate linear codes over the ring $\mathbb{Z}_4 + u\mathbb{Z}_4 + v\mathbb{Z}_4 + w\mathbb{Z}_4 + uv\mathbb{Z}_4 + uw\mathbb{Z}_4 + vw\mathbb{Z}_4 + uvw\mathbb{Z}_4$, with conditions $u^2=u$, $v^2=v$, $w^2=w$, $uv=vu$, $uw=wu$ and…
We present a list decoding algorithm for quaternary negacyclic codes over the Lee metric. To achieve this result, we use a Sudan-Guruswami type list decoding algorithm for Reed-Solomon codes over certain ring alphabets. Our decoding…
Self-dual codes over $\Z_2\times\Z_4$ are subgroups of $\Z_2^\alpha \times\Z_4^\beta$ that are equal to their orthogonal under an inner-product that relates to the binary Hamming scheme. Three types of self-dual codes are defined. For each…
Linear complementary dual codes were defined by Massey in 1992, and were used to give an optimum linear coding solution for the two user binary adder channel. In this paper, we define the analog of LCD codes over fields in the ambient space…
We consider a class (M, g, q) of four-dimensional Riemannian manifolds M, where besides the metric g there is an additional structure q, whose fourth power is the unit matrix. We use the existence of a local coordinate system such that…
The binary Reed-Muller codes can be characterized as the radical powers of a modular algebra. We use the Groebner bases to decode these codes.
In classical coding theory, Gray isometries are usually defined as mappings between finite Frobenius rings, which include the ring $Z_m$ of integers modulo $m$, and the finite fields. In this paper, we derive an isometric mapping from $Z_8$…
The idea of (combinatorial) Gray codes is to list objects in question in such a way that two successive objects differ in some pre-specified small way. In this paper, we utilize beta-description trees to cyclicly Gray code three classes of…
A Type IV-II Z4-code is a self-dual code over Z4 with the property that all Euclidean weights are divisible by eight and all codewords have even Hamming weight. In this paper we use generalized bent functions for a construction of…
A framework of monomial codes is considered, which includes linear codes generated by the evaluation of certain monomials. Polar and Reed-Muller codes are the two best-known representatives of such codes and can be considered as two extreme…
Based on the generic construction of linear codes, we construct linear codes over the ring $\Bbb Z_4$ via posets of the disjoint union of two chains. We determine the Lee weight distributions of the quaternary codes. Moreover, we obtain…
In this article, we construct infinite families of quaternary (that is, over the ring $\mathbb{Z}_4$) $\mathcal{C}_{D}$-codes, where the defining set $D$ is derived utilizing a two-generator simplicial complex, and determine their Lee…
Recursive list decoding is considered for Reed-Muller (RM) codes. The algorithm repeatedly relegates itself to the shorter RM codes by recalculating the posterior probabilities of their symbols. Intermediate decodings are only performed…
In this paper, by treating Reed-Muller (RM) codes as a special class of low-density parity-check (LDPC) codes and assuming that sub-blocks of the parity-check matrix are randomly interleaved to each other as Gallager's codes, we present a…
A linear code of length $n$ over a finite chain ring $R$ with residue field $\F_q$ is a $R$-submodule of $R^n$. A $R$-linear code is a code over $\F_q$ (not necessarily linear) which is the generalized Gray map image of a linear code over…
We determine the weight spectra of the Reed-Muller codes $RM(m-3,m)$ for $m\ge 6$ and $RM(m-4,m)$ for $m\ge 8$. The technique used is induction on $m$, using that the sum of two weights in $RM(r-1,m-1)$ is a weight in $RM(r,m)$, and using…
The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{\alpha_1} \times \mathbb{Z}_4^{\alpha_2} \times \mathbb{Z}_8^{\alpha_3}$, and can be seen as linear codes over $\mathbb{Z}_2$ when…