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]

The MacWilliams' Extension Theorem is a classical result by Florence Jessie MacWilliams. It shows that every linear isometry between linear block-codes endowed with the Hamming distance can be extended to a linear isometry of the ambient space. Such an extension fails to exist in general for rank-metric codes, that is, one can easily find examples of linear isometries between rank-metric codes which cannot be extended to linear isometries of the ambient space. In this paper, we explore to what extent a MacWilliams' Extension Theorem may hold for rank-metric codes. We provide an extensive list of examples of obstructions to the existence of an extension, as well as a positive result. [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]