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In this paper polynomial maps are represented by the use of matrices whose entries are numbered by pair of multiindices and a new product of such matrices is introduced. A matrix representation of composition of polynomial maps is given. In…
We define and study a class of Reed-Muller type error-correcting codes obtained from elementary symmetric functions in finitely many variables. We determine the code parameters and higher weight spectra in the simplest cases.
Cyclic codes are an interesting type of linear codes and have applications in communication and storage systems due to their efficient encoding and decoding algorithms. They have been studied for decades and a lot of progress has been made.…
We apply polynomial techniques (linear programming) to obtain lower and upper bounds on the covering radius of spherical designs as function of their dimension, strength, and cardinality. In terms of inner products we improve the lower…
Polynomial remainder codes are a large class of codes derived from the Chinese remainder theorem that includes Reed-Solomon codes as a special case. In this paper, we revisit these codes and study them more carefully than in previous work.…
In this article, we introduce new scalar products over finite rings via additive isomorphisms. This allows us to define new notions of right (respectively left) orthogonal codes, that are not necessarily linear. This leads to definitions of…
Some linear codes associated to maximal algebraic curves via Feng-Rao construction are investigated. In several case, these codes have better minimum distance with respect to the previously known linear codes with same length and dimension.
In this paper, we consider minimal linear codes in a general construction of linear codes from q-ary functions. First, we give the sufficient and necessary condition for codewords to be minimal. Second, as an application, we present four…
Linear codes are widely studied in coding theory as they have nice applications in distributed storage, combinatorics, lattices, cryptography and so on. Constructing linear codes with desirable properties is an interesting research topic.…
We will show how to obtain a linear code from a configuration of affine lines in general position and a suitable set of rational points. We will also explain a new decoding algorithm based on the configuration, which seems to be quite…
Subspace codes have received an increasing interest recently due to their application in error-correction for random network coding. In particular, cyclic subspace codes are possible candidates for large codes with efficient encoding and…
A codeword is associated to a linearized polynomial. The weight distribution of the codewords is determined as the linearized polynomial varies in a family of fixed degree. There is a corresponding result on Wenger graphs from linearized…
Mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches, challenging established but often heuristic algorithms based on iterative message passing. Concepts from mathematical…
We study additive quaternary codes whose parameters are close to those of the extended cyclic [12; 6; 6]4-code or to the quaternary linear codes generated by the elliptic quadric in PG(3; 4) or its dual. In particular we characterize those…
Techniques for the evaluation of complex polynomials with one and two variables are introduced. Polynomials arise in may areas such as control systems, image and signal processing, coding theory, electrical networks, etc., and their…
Linear codes have been an interesting topic in both theory and practice for many years. In this paper, for an odd prime power $q$, we construct some class of linear code over finite field $\mathbb{F}_q$ with defining set be the preimage of…
Given a quadratic two-parameter matrix polynomial in Newton basis $Q_{N} (\lambda ,\mu)$, we construct a vector space of linear two-parameter matrix polynomials and identify a set of linearizations which lie in the vector space. We also…
This paper investigates the concept of self-dual convolutional code. We derive the basic properties of this interesting class of codes and we show how some of the techniques to construct self-dual linear block codes generalize to self-dual…
Product codes are widespread in optical communications, thanks to their high throughput and good error-correction performance. Systematic polar codes have been recently considered as component codes for product codes. In this paper, we…
A construction is presented that allows to produce subspace codes of long length using subspace codes of shorter length in combination with a rank metric code. The subspace distance of the resulting code, called linkage code, is as good as…