Linearized trinomials with maximum kernel

Journal of algebra combinatorics discrete structures and applications, 2023

In this paper, we are interested in monomial codes with associated vector a = (a0, a1,. .. , an−1), introduced in [4], and more generally in linear codes invariant under a monomial matrix M = diag(a0, a1,. .. , an−1)Pσ where σ is a permutation and Pσ its associated permutation matrix. We discuss some connections between monomial codes and codes invariant under an arbitrary monomial matrix M. Next, we identify monomial codes with associated vector a = (a0, a2,. .. , an−1) by the ideals of the polynomial ring R q,n := F q [x] x n − n−1 i=0 ai , via a special isomorphism ϕ a which preserves the Hamming weight and differs from the classical isomorphism used in the case of cyclic codes and their generalizations. This correspondence leads to some basic characterizations of monomial codes such as generator polynomials, parity check polynomials, and others. Next, we focus on the structure of −quasi-monomial (−QM) codes of length n = m , where on the one hand, we characterize them by the R q,m −submodules of R q,m. On the other hand, −QM codes are seen as additive monomial codes over the extension F q /F q. So, as in the case of quasi-cyclic codes [8], we characterize those codes that have F q −linear images with respect to a basis of the extension F q /F q , based on the CRT decomposition. Finally, we show that −QM codes and additive monomial codes are asymptotically good.

Linear Algebra and its Applications, 2020

Linearized polynomials appear in many different contexts, such as rank metric codes, cryptography and linear sets, and the main issue regards the characterization of the number of roots from their coefficients. Results of this type have been already proved in [7, 10, 24]. In this paper we provide bounds and characterizations on the number of roots of linearized polynomials of this form ax + b 0 x q s + b 1 x q s+n + b 2 x q s+2n +. .. + b t−1 x q s+n(t−1) ∈ F q nt [x], with gcd(s, n) = 1. Also, we characterize the number of roots of such polynomials directly from their coefficients, dealing with matrices which are much smaller than the relative Dickson matrices and the companion matrices used in the previous papers. Furthermore, we develop a method to find explicitly the roots of a such polynomial by finding the roots of a q n-polynomial. Finally, as an applications of the above results, we present a family of linear sets of the projective line whose points have a small spectrum of possible weights, containing most of the known families of scattered linear sets. In particular, we carefully study the linear sets in PG(1, q 6) presented in [9].

Finite Fields and Their Applications, 2004

We generalize a recent idea for constructing codes over a finite field F q by evaluating a certain collection of polynomials over F q at elements of an extension field. We show that many codes with the best parameters presently known can be obtained by this construction. In particular, a new linear code, a ½40; 23; 10-code over F 5 is discovered. Moreover, several families of optimal and near-optimal codes can also be obtained by this method. We call a code near-optimal if its minimum distance is within 1 of the known upper bound.

Finite Fields and Their Applications, 2013

Cyclic, negacyclic and constacyclic codes are part of a larger class of codes called polycyclic codes; namely, those codes which can be viewed as ideals of a factor ring of a polynomial ring. The structure of the ambient ring of polycyclic codes over GR(p a , m) and generating sets for its ideals are considered. It is shown that these generating sets are strong Groebner bases. A method for finding such sets in the case that a = 2 is also given. The Hamming distance of certain constacyclic codes of length ηp s and 2ηp s over Fpm is computed. A method, which determines the Hamming distance of the constacyclic codes of length ηp s and 2ηp s over GR(p a , m), where (η, p) = 1, is described. In particular, the Hamming distance of all cyclic codes of length p s over GR(p 2 , m) and all negacyclic codes of length 2p s over Fpm is determined explicitly.

2002

this paper, we analyse the irreducibility of trinomials defined over F 2 [X].For elliptic curve cryptography, prime extensions fields F 2 p are recommendedto avoid current attacks like the Weil descent attack [3]. One often representsF 2p as a quotient F 2 [X]/(f(X)), where f is an irreducible polynomial over F 2 ofdegree p

In this paper we study polycyclic codes of length $p^{s_1} × · · · × p^{s_n}$ over $F_{p^a}$ generated by a single monomial. These codes form a special class of abelian codes. We show that these codes arise from the product of certain single variable codes and we determine their minimum Hamming distance. Finally we extend the results of Massey et. al. in [10] on the weight retaining property of monomials in one variable to the weight retaining property of monomials in several variables.

2015 IEEE International Symposium on Information Theory (ISIT), 2015

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 decoding algorithms. In this paper we consider such cyclic codes and provide constructions of optimal codes for which their codewords do not have full orbits. We further introduce a new way to represent subspace codes by a class of polynomials called subspace polynomials. We present some constructions of such codes which are cyclic and analyze their parameters.

Series on Coding Theory and Cryptology, 2010

This chapter gives an introduction to algebraic coding theory and a survey of constructions of some of the well known classes of algebraic block codes such as cyclic codes, BCH codes, Reed-Solomon codes, Hamming codes, quadratic residue codes, and quasi-cyclic (QC) codes. It then describes some recent generalizations of QC codes and open problems related to them. Also discussed in this chapter are elementary bounds on the parameters of a linear code, the main problem of algebraic coding theory, and some algebraic and combinatorial methods of obtaining new codes from existing codes. It also includes a section on codes over Z4, integers modulo 4, due to increased attention given to these codes recently. Moreover, a recently created database of best known codes over Z4 is introduced.

Computational & Applied Mathematics, 2011

This paper introduces novel constructions of cyclic codes using semigroup rings instead of polynomial rings. These constructions are applied to define and investigate the BCH, alternant, Goppa, and Srivastava codes. This makes it possible to improve several recent results due to Andrade and Palazzo [1].

Involve, a Journal of Mathematics, 2009

New metrics and distances for linear codes over the ring Fq[u]/(u t ) are defined, which generalize the Gray map, Lee weight, and Bachoc weight; and new bounds on distances are given. Two characterizations of self-dual codes over Fq[u]/(u t ) are determined in terms of linear codes over Fq. An algorithm to produce such self-dual codes is also established. Many optimal codes have been obtained by studying codes over general rings rather than fields. Lately, codes over finite chain rings (of which F q [u]/(u t ) is an example) have been a source of many interesting properties . Gulliver and Harada [4] found good examples of ternary codes over F 3 using a particular type of Gray map. Siap and Ray-Chaudhuri in established a relation between codes over F q [u]/(u 2 -a) and codes over F q , which was used to obtained new codes over F 3 and F 5 . In this paper we present a certain generalization of the method used in [4] and , defining a family of metrics for linear codes over F q [u]/(u t ) and obtaining as particular examples the Gray map, the Gray weight, the Lee weight and the Bachoc weight. For the latter, we give a new bound on the distance of those codes. It also shows that the Gray images of codes over F 2 + uF 2 are more powerful than codes obtained by the so-called u-u+v condition. With these tools in hand, we study conditions for self-duality of codes over F q [u]/(u t ). In the authors study the case of self-dual cyclic codes in terms of the generator polynomials. In this paper we study self-dual codes in terms of linear codes over F q that are obtained as images under the maps defined on the first part of the paper. We provide a way to construct many self-dual codes over F q starting from a self-dual code over F q [u]/(u t ). We also study self-dual codes in terms of the torsion codes, and provide a way to construct many self-dual codes over F q [u]/(u t ) starting from a self-orthogonal code over F q . Our results contain many of the properties studied by Bachoc for self-dual codes over 2. Metric for Codes over F q [u]/(u t ). We will use R(q, t) to denote the commutative ring F q [u]/(u t ). The q t elements of this ring can be represented in two different forms, and we will use the most appropriate in each case. First, we can use the * The project was partially supported by Office of Research of the University of Michigan-Flint.