Search

In this paper, we study constacyclic codes of length n = 7 p s over a finite field of characteristics p , where p ≠ 7 is an odd prime number and s a positive integer. The previous methods in the literature that were used to compute the Hamming distances of repeated-root constacyclic codes of lengths n p s with 1 ≤ n ≤ 6 cannot be applied to completely determine the Hamming distances of those with n = 7. This is due to the high computational complexity involved and the large number of unexpected intermediate results that arise during the computation. To overcome this challenge, we propose a computer-assisted method for determining the Hamming distances of simple-root constacyclic codes of length 7, and then utilize it to derive the Hamming distances of the repeated-root constacyclic codes of length 7 p s. Our method is not only straightforward to implement but also efficient, making it applicable to these codes with larger values of n as well. In addition, all self-orthogonal, dual-containing, self-dual, MDS and AMDS codes among them will also be characterized. [ABSTRACT FROM AUTHOR]

We consider sparse polynomials in N variables over a finite field, and ask whether they vanish on a set S N , where S is a set of nonzero elements of the field. We see that if for a polynomial f , there is ▪ with ▪, then there is such a ▪ in every ball (with respect to the Hamming distance) inside S N , where the radius of the ball is bounded by a multiple of the logarithm of the number of monomials that appear in f. A similar result holds for the solutions of the equations f 1 = ⋯ = f r = 0 inside S N. When 0 ∉ S , this provides algorithms for testing whether a given polynomial f vanishes on S N and for checking whether f 1 = ⋯ = f r = 0 has a solution in S N that are of time complexity in O (n c log ⁡ (n)) , where c > 0 and n is the size of the input polynomials given in sparse representation. • We check whether a polynomial over a finite field vanishes on a rectangular domain. • We show that nonzeros of a polynomial f can be found inside balls with radius logarithmic in the number of monomials of f. • We explain how to find solutions of a polynomial system inside rectangular domains in quasi-polynomial time. [ABSTRACT FROM AUTHOR]

Multi-twisted (MT) additive codes over finite fields constitute an important class of additive codes and are generalizations of cyclic and constacyclic additive codes. In this paper, we employ probabilistic methods and results from groups and geometry to study the asymptotic behavior of the rates and relative Hamming distances of two special subclasses of MT additive codes over finite fields, viz. MT additive self-orthogonal codes and MT additive codes with complementary duals (MT ACD codes) with respect to ordinary, Hermitian, and ⁎ trace bilinear forms. More precisely, let p be an odd prime, q be a prime power coprime to p , v be the multiplicative order of q modulo p , and let η be the largest positive integer such that p η | (q v − 1). Here we establish the existence of asymptotically good infinite sequences of MT additive self-orthogonal and ACD codes of length p α ℓ and block length p α over F q t with p α → ∞ , relative Hamming distance at least δ and rates v p η ℓ t and 2 v p η ℓ t with respect to the aforementioned trace bilinear forms, where ℓ ≥ 1 , t ≥ 2 are integers, F q t is the finite field of order q t and δ is a positive real number satisfying h q t (δ) < 1 2 − 1 2 ℓ t , (here h q t is the q t -ary entropy function). This shows that MT additive self-orthogonal and ACD codes over finite fields are asymptotically good. As a special case, we deduce that constacyclic additive self-orthogonal and ACD codes over finite fields are asymptotically good. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of linear codes over finite fields, viz. skew multi-twisted codes (or skew generalized quasi-twisted codes), which is a generalization of some well-known classes of linear codes such as cyclic codes and quasi-cyclic codes. We also study algebraic structures of skew multi-twisted codes and their Galois duals (i.e., orthogonal complements with respect to the Galois inner product). We also view skew multi-twisted codes as direct sums of certain concatenated codes, which gives rise to a method to construct these codes. We also determine a lower bound on their minimum Hamming distances using their multilevel concatenated structure. Apart from this, we determine the parity-check polynomial of each skew multi-twisted code, and obtain BCH type bounds on their minimum Hamming distances. We also determine generating sets of Galois duals of some skew multi-twisted codes from generating sets of these codes. We also derive necessary and sufficient conditions under which a skew multi-twisted code is (i) Galois self-dual, (ii) Galois self-orthogonal and (iii) Galois LCD (linear with complementary dual). As an application of this result, we also provide enumeration formulae for all Hermitian self-dual, Hermitian self-orthogonal, Hermitian LCD and Euclidean LCD multi-twisted codes over finite fields. We also obtain many linear codes with best known and optimal parameters from 1-generator skew multi-twisted codes over finite fields F 8 and F 9. [ABSTRACT FROM AUTHOR]

Let p be prime integer, s and m positive integers; f (x) be a monic irreducible polynomial over R where R : = F p m [ u ] / 〈 u 2 〉. This paper gives a complete classification of polycyclic codes over R associated to f (x) p s , and the other hand, when f (x) ∈ F p m [ x ] , their Hamming distances and their annihilators are explicitly given. [ABSTRACT FROM AUTHOR]

Let F be any field and A 1 , … , A m be finite subsets of F . We determine the maximum number of common zeroes a linearly independent family of r polynomials of degree at most d of F [ x 1 , … , x m ] can have in A 1 × … × A m . In the case when F is a finite field, our results resolve the problem of determining the generalized Hamming weights of affine Cartesian codes. This is a generalization of the work of Heijnen and Pellikaan where these were determined for the generalized Reed–Muller codes. Finally, we determine the duals of affine Cartesian codes and compute their generalized Hamming weights as well. [ABSTRACT FROM AUTHOR]

Let F q denote the finite field of order q , let m 1 , m 2 , ⋯ , m ℓ be positive integers satisfying gcd ⁡ ( m i , q ) = 1 for 1 ≤ i ≤ ℓ , and let n = m 1 + m 2 + ⋯ + m ℓ . Let Λ = ( λ 1 , λ 2 , ⋯ , λ ℓ ) be fixed, where λ 1 , λ 2 , ⋯ , λ ℓ are non-zero elements of F q . In this paper, we study the algebraic structure of Λ-multi-twisted codes of length n over F q and their dual codes with respect to the standard inner product on F q n . We provide necessary and sufficient conditions for the existence of a self-dual Λ-multi-twisted code of length n over F q , and obtain enumeration formulae for all self-dual and self-orthogonal Λ-multi-twisted codes of length n over F q . We also derive some sufficient conditions under which a Λ-multi-twisted code is linear with complementary dual (LCD). We determine the parity-check polynomial of all Λ-multi-twisted codes of length n over F q and obtain a BCH type bound on their minimum Hamming distances. We also determine generating sets of dual codes of some Λ-multi-twisted codes of length n over F q from the generating sets of the codes. Besides this, we provide a trace description for all Λ-multi-twisted codes of length n over F q by viewing these codes as direct sums of certain concatenated codes, which leads to a method to construct these codes. We also obtain a lower bound on their minimum Hamming distances using their multilevel concatenated structure. [ABSTRACT FROM AUTHOR]

The use of skew polynomial rings allows to endow linear codes with cyclic structures which are not cyclic in the classical (commutative) sense. Whenever these skew cyclic structures are carefully chosen, some control over the Hamming distance is gained, and it is possible to design efficient decoding algorithms. In this paper, we give a version of the Hartmann–Tzeng bound that works for a wide class of skew cyclic codes. We also provide a practical method for constructing them with designed distance. For skew BCH codes, which are covered by our constructions, we discuss decoding algorithms. Detailed examples illustrate both the theory as the constructive methods it supports. [ABSTRACT FROM AUTHOR]

The structure of λ -constacyclic codes of length 2 s over the Galois ring GR ( 2 a , m ) is obtained, for any unit λ of the form 4 z − 1 , z ∈ GR ( 2 a , m ) . The duals codes and necessary and sufficient conditions for the existence of a self-dual λ -constacyclic code are provided. Among others, this structure is used to establish the Hamming, homogeneous, and Rosenbloom–Tsfasman distances, and Rosenbloom–Tsfasman weight distribution of all such constacyclic codes. [ABSTRACT FROM AUTHOR]

Let p ≠ 3 be any prime and l ≠ 3 be any odd prime with gcd ⁡ ( p , l ) = 1 . The multiplicative group F q ⁎ = 〈 ξ 〉 can be decomposed into mutually disjoint union of gcd ⁡ ( q − 1 , 3 l p s ) cosets over the subgroup 〈 ξ 3 l p s 〉 , where ξ is a primitive ( q − 1 ) th root of unity. We classify all repeated-root constacyclic codes of length 3 l p s over the finite field F q into some equivalence classes by this decomposition, where q = p m , s and m are positive integers. According to these equivalence classes, we explicitly determine the generator polynomials of all repeated-root constacyclic codes of length 3 l p s over F q and their dual codes. Self-dual cyclic codes of length 3 l p s over F q exist only when p = 2 . We give all self-dual cyclic codes of length 3 ⋅ 2 s l over F 2 m and their enumeration. We also determine the minimum Hamming distance of these codes when gcd ⁡ ( 3 , q − 1 ) = 1 and l = 1 . [ABSTRACT FROM AUTHOR]

Let F q be a finite field of q elements with q ≡ 3 ( mod 8 ) and p | q − 1 , where p is an odd prime. In this paper, we use matrix method to give all primitive idempotents in two rings F q [ x ] / 〈 x 4 p n − 1 〉 and F q [ x ] / 〈 x 8 p n − 1 〉 . Furthermore, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length 4 p n and 8 p n over F q . [ABSTRACT FROM AUTHOR]

Let p be a prime number. In this paper, we study cyclic codes over the ring Z p [ u , v ] / 〈 u 2 , v 2 , u v − v u 〉 . We find a unique set of generators for these codes. We also study the rank and the Hamming distance of these codes. We obtain all except one ternary optimal code of length 12 as the Gray image of the cyclic codes over the ring Z p [ u , v ] / 〈 u 2 , v 2 , u v − v u 〉 . [ABSTRACT FROM AUTHOR]

Let F q be a finite field with q elements and n = l 1 m 1 l 2 m 2 , m 1 ≥ 1 , m 2 ≥ 1 , where l 1 , l 2 are distinct primes and l 1 l 2 | q − 1 . In this paper, we give all irreducible factors of x l 1 m 1 l 2 m 2 − 1 over F q and all primitive idempotents in a ring F q [ x ] / 〈 x l 1 m 1 l 2 m 2 − 1 〉 . Furthermore, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length l 1 m 1 l 2 m 2 over F q . [ABSTRACT FROM AUTHOR]

Abstract: By regarding a finite field as a vector space over the prime field with a basis consisting of powers of an element, a Hamming distance is defined on the finite field with respect to the power basis. We consider the existence of isometric homomorphisms between such finite fields, and characterize the isometric embeddings for even characteristic by arithmetical conditions. Moreover, a canonical Hamming metric is defined in a certain infinite dimensional algebraic extension of a finite field. [Copyright &y& Elsevier]

In this paper we introduce the notion of λ -constacyclic codes over finite rings R for arbitrary element λ of R. We study the non-invertible-element constacyclic codes (NIE-constacyclic codes) over finite principal ideal rings (PIRs). We determine the algebraic structures of all NIE-constacyclic codes over finite chain rings, give the unique form of the sets of the defining polynomials and obtain their minimum Hamming distances. A general form of the duals of NIE-constacyclic codes over finite chain rings is also provided. In particular, we give a necessary and sufficient condition for the dual of an NIE-constacyclic code to be an NIE-constacyclic code. Using the Chinese Remainder Theorem, we study the NIE-constacyclic codes over finite PIRs. Furthermore, we construct some optimal NIE-constacyclic codes over finite PIRs in the sense that they achieve the maximum possible minimum Hamming distances for some given lengths and cardinalities. [ABSTRACT FROM AUTHOR]

Abstract: We investigate negacyclic and cyclic codes of length over the finite field . Negacyclic codes of length are precisely the ideals of the chain ring . This structure is then used to obtain the Hamming distance distribution of the class of such negacyclic codes, which also provides Hamming weight distributions and enumerations of several codes. An one-to-one correspondence between negacyclic and cyclic codes is established to carry accordingly those results of negacyclic codes to cyclic codes. [Copyright &y& Elsevier]

Let p ≠ 3 be a prime, s , m be positive integers, and λ be a nonzero element of the finite field F p m . In [22] and [20] , when the generator polynomials have one or two different irreducible factors, the Hamming distances of λ -constacyclic codes of length 3 p s over F p m have been considered. In this paper, we obtain that the Hamming distances of the repeated-root λ -constacyclic codes of length l p s can be determined by the Hamming distances of the simple-root γ -constacyclic codes of length l , where l is a positive integer and λ = γ p s . Based on this result, the Hamming distances of the repeated-root λ -constacyclic codes of length 3 p s are given when the generator polynomials have three different irreducible factors. Hence, the Hamming distances of all such constacyclic codes are determined. As an application, we obtain all optimal λ -constacyclic codes of length 3 p s with respect to the Griesmer bound and the Singleton bound. Among others, several examples show that some of our codes have the best known parameters with respect to the code tables in [19]. [ABSTRACT FROM AUTHOR]

In the last decade there has been a great interest in extending results for codes equipped with the Hamming metric to analogous results for codes endowed with the rank metric. This work follows this thread of research and studies the characterization of systematic generator matrices (encoders) of codes with maximum rank distance. In the context of Hamming distance these codes are the so-called Maximum Distance Separable (MDS) codes and systematic encoders have been fully investigated. In this paper we investigate the algebraic properties and representation of encoders in systematic form of Maximum Rank Distance (MRD) codes and Maximum Sum Rank Distance (MSRD) codes. We address both block codes and convolutional codes separately and present necessary and sufficient conditions for an encoder in systematic form to generate a code with maximum (sum) rank distance. These characterizations are given in terms of certain matrices that must be superregular in a extension field and that preserve superregularity after some transformations performed over the base field. We conclude the work presenting some examples of Maximum Sum Rank convolutional codes over small fields. For the given parameters the examples obtained are over smaller fields than the examples obtained by other authors. [ABSTRACT FROM AUTHOR]

We consider a generalisation of a conjecture by Patterson and Wiedemann from 1983 on the Hamming distance of a function from F q n to F q to the set of affine functions from F q n to F q. We prove the conjecture for each q such that the characteristic of F q lies in a subset of the primes with density 1 and we prove the conjecture for all q by assuming the generalised Riemann hypothesis. Roughly speaking, we show the existence of functions for which the distance to the affine functions is maximised when n tends to infinity. This also determines the asymptotic behaviour of the covering radius of the [ q n , n + 1 ] Reed-Muller code over F q and so answers a question raised by Leducq in 2013. Our results extend the case q = 2 , which was recently proved by the author and which corresponds to the original conjecture by Patterson and Wiedemann. Our proof combines evaluations of Gauss sums in the semiprimitive case, probabilistic arguments, and methods from discrepancy theory. [ABSTRACT FROM AUTHOR]