Cocyclic butson Hadamard matrices and codes over Zn via the trace map

Over the past couple of years, trace maps over Galois fields and Galois rings have been used very succesfully o construct cocyclic Hadamard, complex Hadamard and Butson Hadamard matrices and subsequently to generate simplex codes over Z4, Z2 and ZP and new linear codes over ZP. Here we define a new map, the trace-like map and more generally the weighted map and extend these techniques to construct cocyclic Budson Hadamard matrices of order (nm) for all n and m and linear and non-linear codes over Zn

Valued rank-metric codes

In this paper, we study linear spaces of matrices defined over discretely valued fields and discuss their dimension and minimal rank drops over the associated residue fields. To this end, we take first steps into the theory of rank-metric codes over discrete valuation rings by means of skew algebras derived from Galois extensions of rings. Additionally, we model projectivizations of rank-metric codes via Mustafin varieties, which we then employ to give sufficient conditions for a decrease in the dimension.Comment: 33 page

Properties of trace maps and their applications to coding theory

In this thesis we study the application of trace maps over Galois fields and Galois rings in the construction of non-binary linear and non-linear codes and mutually unbiased bases. Properties of the trace map over the Galois fields and Galois rings has been used very successfully in the construction of cocyclic Hadamard, complex Hadamard and Butson Hadamard matrices and consequently to construct linear codes over integers modulo prime and prime powers. These results provide motivation to extend this work to construct codes over integers modulo . The prime factorization of integers paved the way to focus our attention on the direct product of Galois rings and Galois fields of the same degree. We define a new map over the direct product of Galois rings and Galois fields by using the usual trace maps. We study the fundamental properties of the this map and notice that these are very similar to that of the trace map over Galois rings and Galois fields. As such this map called the trace-like map and is used to construct cocyclic Butson Hadamard matrices and consequently to construct linear codes over integers modulo . We notice that the codes construct in this way over the integers modulo 6 is simplex code of type . A further generalization of the trace-like map called the weighted-trace map is defined over the direct product of Galois rings and Galois fields of different degrees. We use the weighted-trace map to construct some non-linear codes and mutually unbiased bases of odd integer dimensions. Further more we study the distribution of over the Galois fields of degree 2 and use it to construct 2-dimensional, two-weight, self-orthogonal codes and constant weight codes over integers modulo prime

STBCs from Representation of Extended Clifford Algebras

A set of sufficient conditions to construct $\lambda$-real symbol Maximum Likelihood (ML) decodable STBCs have recently been provided by Karmakar et al. STBCs satisfying these sufficient conditions were named as Clifford Unitary Weight (CUW) codes. In this paper, the maximal rate (as measured in complex symbols per channel use) of CUW codes for $\lambda=2^a,a\in\mathbb{N}$ is obtained using tools from representation theory. Two algebraic constructions of codes achieving this maximal rate are also provided. One of the constructions is obtained using linear representation of finite groups whereas the other construction is based on the concept of right module algebra over non-commutative rings. To the knowledge of the authors, this is the first paper in which matrices over non-commutative rings is used to construct STBCs. An algebraic explanation is provided for the 'ABBA' construction first proposed by Tirkkonen et al and the tensor product construction proposed by Karmakar et al. Furthermore, it is established that the 4 transmit antenna STBC originally proposed by Tirkkonen et al based on the ABBA construction is actually a single complex symbol ML decodable code if the design variables are permuted and signal sets of appropriate dimensions are chosen.Comment: 5 pages, no figures, To appear in Proceedings of IEEE ISIT 2007, Nice, Franc

Editor’s Note. Special Issue Algebraic Coding Theory: New Trends and Its Connections

Dear Colleagues The purpose of this special issue of Journal of Algebra,Combinatorics, Discrete Structures and Applications was to collect a sample of papers in active areas of research in algebraic coding theory and its connections to other areas. A number of researchers submitted manuscripts to the special issue. After a thorough review process, six articles have been selected to appear in the special issue. We thank all researchers who submitted an article. Their contributions are sincerely appreciated, regardless of whether they have been accepted for publication or not. We are particularly grateful to our small number of dedicated reviewers who did a meticulous job of reviewing in a short period of time. The articles selected for this special issue are a representative sample of the current research trends in algebraic coding theory. In their article "Construction of quasi-twisted codes and enumeration of defining polynomials", Gulliver and Venkaiah enumerate all twistulant matrices of a given size and use that information to construct quasi-twisted (QT) codes with better parameters and they start new databases over $GF(17)$ and $GF(19)$. QT codes have been studied extensively in coding theory and they continue to yield useful results. In the article "Locally recoverable codes from planar graphs" Haymaker and O’Pella construct codes that are locally recoverable from 3-connected regular and almost regular graphs. Furthermore, they present methods of constructing regular and almost regular planar graphs. In the paper "Constructions of MDS convolutional codes using superregular matrices", Lieb and Pinto show how to obtain MDS convolutional codes from superregular matrices with certain properties. They provide explicit ways of constructing generator matrices of MDS convolutional codes from superregular matrices. In the paper titled "G-codes over formal power series rings", Korban et al. introduce G-codes over an infinite ring, using tools from group rings. They study the duality properties of these codes and show that their projections are G-codes over finite chain rings. They prove similar results for the lifts of codes over finite chain rings as well. In "$Z_q(Z_q+uZ_q)$-linear skew constacyclic codes", Melakhessou et al. consider Zq(Zq+uZq) skew constacyclic codes where q is a prime power and $u^2=0$. They describe the generator polynomials, the minimal spanning sets, and sizes of these codes. They also obtain some new $Z_4$-codes from the Gray images of these codes. In "Weight distributions of some constacyclic codes over a finite field and isodual constacyclic codes", Singh describes the weight distribution of a family of constacyclic codes over $F_q$. Singh also constructs a family of non-binary isodual-constacyclic codes of a special length and gives specific examples of the constructions. Algebraic Coding Theory continues to be an active area of research with many theoretical and applied aspects. We believe that this special issue will help disseminate recent results to a broad audience in an open access journal and promote further research

Quasi self-dual codes over non-unital rings from three-class association schemes

Let $E$ and $I$ denote the two non-unital rings of order 4 in the notation of (Fine, 93) defined by generators and relations as $E = \langle a, b \mid 2a = 2b = 0, a^2 = a, b^2 = b, ab = a, ba = b\rangle$ and $I = \langle a, b \mid 2a = 2b = 0, a^2 = b, ab = 0\rangle$. Recently, Alahmadi et al classified quasi self-dual (QSD) codes over the rings $E$ and $I$ for lengths up to 12 and 6, respectively. The codes had minimum distance at most 2 in the case of $I$, and 4 in the case of $E$. In this paper, we present two methods for constructing linear codes over these two rings using the adjacency matrices of three-class association schemes. We show that under certain conditions the constructions yield QSD or Type Ⅳ codes. Many codes with minimum distance exceeding 4 are presented. The form of the generator matrices of the codes with these constructions prompted some new results on free codes over $E$ and $I$