Related papers: Another Generalization of the Reed-Muller Codes
Flag codes that are orbits of a cyclic subgroup of the general linear group acting on flags of a vector space over a finite field, are called cyclic orbit flag codes. In this paper we present a new contribution to the study of such codes…
In this paper, we study the problem of constructing projective systematic authentication schemes based on binary linear codes. In systematic authentication, a tag for authentication is generated and then appended to the information, also…
In this article we count the number of generalized Reed-Solomon (GRS) codes of dimension k and length n, including the codes coming from a non-degenerate conic plus nucleus. We compare our results with known formulae for the number of…
Duadic group algebra codes are a generalization of quadratic residue codes. This paper settles an open problem raised by Zhu concerning the existence of duadic group algebra codes. These codes can be used to construct degenerate quantum…
Kitaev's toric code is constructed using a finite gauge group from gauge theory. Such gauge theories can be generalized with the gauge group generalized to any finite-dimensional semisimple Hopf algebra. This also leads to generalizations…
Quasi-cyclic codes form an important class of algebraic codes that includes cyclic codes as a special subclass. This chapter focuses on the algebraic structure of quasi-cyclic codes, first. Based on these structural properties, some…
As we all know, many interesting and important codes are obtained by modifying or combining existing codes. In this paper, we focus on generalized Roth-Lempel (in short, GRL) codes and define a class of extended codes, i.e., the extended…
The hull of a linear code is defined to be the intersection of the code and its dual. When the size of the hull is small, it has been proved that some algorithms for checking permutation equivalence of two linear codes and computing the…
We prove the cyclic sum formulas for certain two-parameter multiple series. These are new and non-trivial generalizations of the cyclic sum formulas for multiple zeta values and multiple zeta-star values.
Multivariate multiplicity codes (Kopparty, Saraf, and Yekhanin, J. ACM 2014) are linear codes where the codewords are described by evaluations of multivariate polynomials (with a degree bound) and their derivatives up to a fixed order, on a…
Rewriting is a formalism widely used in computer science and mathematical logic. The classical formalism has been extended, in the context of functional languages, with an order over the rules and, in the context of rewrite based languages,…
Let $G$ be a connected graph and let $\mathbb{X}$ be the set of projective points defined by the column vectors of the incidence matrix of $G$ over a field $K$ of any characteristic. We determine the generalized Hamming weights of the…
The permutation groups of cyclic codes are widely applicable in determining the weight distribution of codes, decoding theory and various other areas. In this paper, by employing two distinct matrix representations, we can relate cyclic…
First- and second-order Reed-Muller (RM(1) and RM(2), respectively) codes are two fundamental error-correcting codes which arise in communication as well as in probabilistically-checkable proofs and learning. In this paper, we take the…
Reed-Muller codes are some of the oldest and most widely studied error-correcting codes, of interest for both their algebraic structure as well as their many algorithmic properties. A recent beautiful result of Saptharishi, Shpilka and Volk…
We show that Reed-Muller codes achieve capacity under maximum a posteriori bit decoding for transmission over the binary erasure channel for all rates $0 < R < 1$. The proof is generic and applies to other codes with sufficient amount of…
In this paper, we study some new factorizations of period-doubling sequences over a $k$-letter alphabet, where $k\geq 2$. First, we define the combinatorial and arithmetic properties of these sequences. Then, we define the kernel words of…
Linear quasi-cyclic product codes over finite fields are investigated. Given the generating set in the form of a reduced Gr{\"o}bner basis of a quasi-cyclic component code and the generator polynomial of a second cyclic component code, an…
First we generalize the Thue-Morse sequence (the generalized Thue-Morse sequences) by a cyclic permutations and p-adic system, and consider the necessary-sufficient condition that it is non-periodic. Moreover if the generalized Thue-Morse…
This article introduces Generalized Hyperderivative Reed-Solomon codes (GHRS codes), which generalize NRT Reed-Solomon codes. Its main results are as follows: 1) every GHRS code is MDS, 2) the dual of a GHRS code is also an GHRS code, 3)…