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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]

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 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]

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 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]

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]

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]