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

Lattices from Codes for Harnessing Interference: An Overview and Generalizations

In this paper, using compute-and-forward as an example, we provide an overview of constructions of lattices from codes that possess the right algebraic structures for harnessing interference. This includes Construction A, Construction D, and Construction $\pi_A$ (previously called product construction) recently proposed by the authors. We then discuss two generalizations where the first one is a general construction of lattices named Construction $\pi_D$ subsuming the above three constructions as special cases and the second one is to go beyond principal ideal domains and build lattices over algebraic integers

Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page

Reliable Physical Layer Network Coding

When two or more users in a wireless network transmit simultaneously, their electromagnetic signals are linearly superimposed on the channel. As a result, a receiver that is interested in one of these signals sees the others as unwanted interference. This property of the wireless medium is typically viewed as a hindrance to reliable communication over a network. However, using a recently developed coding strategy, interference can in fact be harnessed for network coding. In a wired network, (linear) network coding refers to each intermediate node taking its received packets, computing a linear combination over a finite field, and forwarding the outcome towards the destinations. Then, given an appropriate set of linear combinations, a destination can solve for its desired packets. For certain topologies, this strategy can attain significantly higher throughputs over routing-based strategies. Reliable physical layer network coding takes this idea one step further: using judiciously chosen linear error-correcting codes, intermediate nodes in a wireless network can directly recover linear combinations of the packets from the observed noisy superpositions of transmitted signals. Starting with some simple examples, this survey explores the core ideas behind this new technique and the possibilities it offers for communication over interference-limited wireless networks.Comment: 19 pages, 14 figures, survey paper to appear in Proceedings of the IEE

Fast Erasure-and-Error Decoding and Systematic Encoding of a Class of Affine Variety Codes

In this paper, a lemma in algebraic coding theory is established, which is frequently appeared in the encoding and decoding for algebraic codes such as Reed-Solomon codes and algebraic geometry codes. This lemma states that two vector spaces, one corresponds to information symbols and the other is indexed by the support of Grobner basis, are canonically isomorphic, and moreover, the isomorphism is given by the extension through linear feedback shift registers from Grobner basis and discrete Fourier transforms. Next, the lemma is applied to fast unified system of encoding and decoding erasures and errors in a certain class of affine variety codes.Comment: 6 pages, 2 columns, presented at The 34th Symposium on Information Theory and Its Applications (SITA2011

Lattice polytopes in coding theory

In this paper we discuss combinatorial questions about lattice polytopes motivated by recent results on minimum distance estimation for toric codes. We also prove a new inductive bound for the minimum distance of generalized toric codes. As an application, we give new formulas for the minimum distance of generalized toric codes for special lattice point configurations.Comment: 11 pages, 3 figure