A class of almost MDS codes

MDS codes and almost MDS (AMDS) codes are special classes of linear codes, and have important applications in communications, data storage, combinatorial theory, and secrete sharing. The objective of this paper is to present a class of AMDS codes from some BCH codes and determine their parameters. It turns out the proposed AMDS codes are distance-optimal and dimension-optimal locally repairable codes. The parameters of the duals of this class of AMDS codes are also discussed.     

Algebraic-geometry codes, one-point codes, and evaluation codes

One-point codes are those algebraic-geometry codes for which the associated divisor is a non-negative multiple of a single point. Evaluation codes were defined in order to give an algebraic generalization of both one-point algebraic-geometry codes and Reed–Muller codes. Given an -algebra A, an order function on A and given a surjective -morphism of algebras , the ith evaluation code with respect to is defined as the code . In this work it is shown that under a certain hypothesis on the -algebra A, not only any evaluation code is a one-point code, but any sequence of evaluation codes is a sequence of one-point codes. This hypothesis on A is that its field of fractions is a function field over and that A is integrally closed. Moreover, we see that a sequence of algebraic-geometry codes G _i with associated divisors is the sequence of evaluation codes associated to some -algebra A, some order function and some surjective morphism with if and only if it is a sequence of one-point codes.      

Binary Quasi-Cyclic Goppa Codes

We obtain here a necessary and sufficient condition for a certain class of binary Goppa code to be quasi-cyclic. We also give another sufficient condition which is easier to check. We define a class of quasi-cyclic Goppa codes. We find the true dimension for a part of those quasi-cyclic codes. and also a class of extended quasi-cyclic codes the minimum distance of which is equal to the designed distance.     

New optimal subsystem codes

Subsystem codes (also known as operator quantum error-correcting codes) are a generalization of noiseless subsystems, decoherence-free subspaces, and quantum error-correcting codes. In this note, we present a construction for new subsystem codes with parameters _{[[q2+1,(q−1)2,4,q−1]]q}${\left[[q^2+1,{(q-1)}^2,4,q-1]\right]}_q$, where q=2^m$q=2^m$, and m≥1$m\ge 1$ is a positive integer, whose parameters are not covered by the codes available in the literature. Moreover, the constructed subsystem codes are optimal.