Related papers: Good iso-dual AG-codes from towers of function fie…

We give a general recipe for explicitly constructing asymptotically optimal towers of modular curves such as {X_0(l^n): n=1,2,3,...}. We illustrate the method by giving equations for eight towers with various geometric features. We conclude…

In this work, we define composite matrices which are derived from group rings. We extend the idea of G-codes to composite G-codes. We show that these codes are ideals in a group ring, where the ring is a finite commutative Frobenius ring…

Let $G$ be a finite simple graph with edge ideal $I(G)$. Let $J(G)$ denote the Alexander dual of $I(G)$. We show that a description of all induced cycles of odd length in $G$ is encoded in the associated primes of $J(G)^2$. This result…

Let R be a commutative ring with unit and let I be an ideal generated by a regular sequence. Then it is known that the natural sequences 0-> Tor_*^R(R/I,I^s)-> Tor_*^R(R/I,I^s/I^{s+1})-> Tor_{*-1}^R(R/I,I^{s+1})-> 0 are short exact…

This paper investigates the application of the theoretical algebraic notion of a separable ring extension, in the realm of cyclic convolutional codes or, more generally, ideal codes. We work under very mild conditions, that cover all…

Certain towers of function fields with complete splitting of rational places at each stage are constructed. Also, families oof towers with positive N/g ratios are described.

We investigate group coding for arbitrary finite groups acting linearly on a vector space. These yield robust codes based on real or complex matrix groups. We give necessary and sufficient conditions for correct subgroup decoding using…

The interpolation-based decoding that was developed for general evaluation AG codes is shown to be equally applicable to general differential AG codes. A performance analysis of the decoding algorithm, which is parallel to that of its…

This paper proposes some efficient and accurate adaptive two-grid (ATG) finite element algorithms for linear and nonlinear partial differential equations (PDEs). The main idea of these algorithms is to utilize the solutions on the $k$-th…

Affine Grassmann codes are a variant of generalized Reed-Muller codes and are closely related to Grassmann codes. These codes were introduced in a recent work [2]. Here we consider, more generally, affine Grassmann codes of a given level.…

Let $G$ be a graph of minimum degree at least two with no induced subgraph isomorphic to $K_{1,6}$. We prove that if $G$ is not isomorphic to one of eight exceptional graphs, then it is possible to assign two-element subsets of…

This paper is mainly a semi-tutorial introduction to elementary algebraic topology and its applications to Ising-type models of statistical physics, using graphical models of linear and group codes. It contains new material on systematic…

Asymptotically good sequences of ramp secret sharing schemes were given in [Asymptotically good ramp secret sharing schemes, arXiv:1502.05507] by using one-point algebraic geometric codes defined from asymptotically good towers of function…

In this paper, we give algorithms and methods of construction of self-dual codes over finite fields using orthogonal matrices. Randomization in the orthogonal group, and code extension are the main tools. Some optimal, almost MDS, and MDS…

A surprising connection between N=2 gauge theory instanton partition functions and conformal blocks has been recently proposed. We illustrate through simple examples the generalization to asymptotically free N=2 gauge theories

In this paper, several conjectures proposed in [2] are studied, involving the equivalence and duality of polycyclic codes associated with trinomials. According to the results, we give methods to construct isodual and self-dual polycyclic…

In 2002, M. A. Tsfasman and S. G. Vl\u{a}du\c{t} formulated the generalized Brauer-Siegel conjecture for asymptotically exact families of number fields. In this article, we establish this conjecture for asymptotically good towers and…

Let G be a finite abelian group and F a field such that char(F) does not divide |G|. Denote by FG the group algebra of G over F. A (semisimple) abelian code is an ideal of FG. Two codes I and J of FG are G-equivalent if there exists an…

We give lower bounds on the number of effective divisors of degree $\leq g-1$ with respect to the number of places of certain degrees of an algebraic function field of genus $g$ defined over a finite field. We deduce lower bounds and…

In this paper we consider algebraic geometry (AG) codes: a class of codes constructed from algebraic codes (equivalently, using function fields) by Goppa. These codes can be list-decoded using the famous Guruswami-Sudan (GS) list-decoder,…