We study additive codes with 1-rank hulls and examine their properties for various dualities of the finite field of order 4. We give several constructions of additive and linear codes with 1-rank hulls. We also relate...
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We study additive codes with 1-rank hulls and examine their properties for various dualities of the finite field of order 4. We give several constructions of additive and linear codes with 1-rank hulls. We also relate these codes to additive complementary dual codes (ACD). We give an interesting non-existence result for additive codes with a 1-rank hull for the duality M2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_2$$\end{document} in terms of the parity of the number of generators. We conclude by giving substantive computations finding codes with one-rank hulls for small lengths using our results.
Let RSk(a) be a k-dimensional Reed-Solomon (RS) code over F-q associated with a = (alpha(1), ..., alpha(n)) and let h = Pi(n)(i=1)(z - alpha(i)) be a polynomial in variable z. In this paper, by expressing RSk(a) as an...
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Let RSk(a) be a k-dimensional Reed-Solomon (RS) code over F-q associated with a = (alpha(1), ..., alpha(n)) and let h = Pi(n)(i=1)(z - alpha(i)) be a polynomial in variable z. In this paper, by expressing RSk(a) as an L-construction algebraic geometry code, we completely determine the dimension of the hull RSk(a) boolean AND RSk(a)(perpendicular to) in terms of the degree of the derivative of h and some relevant polynomials. As applications, we explicitly determine the parameters of MDS entanglement-assisted quantum error-correcting codes constructed from RS codes, and all linear complementary dual (resp. self-dual) RS codes are also fully described.
The hull of a linear code over finite fields is the intersection of the code and its dual, which was introduced by Assmus and Key. In this paper, we develop a method to construct linear codes with trivial hull (LCD co...
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The hull of a linear code over finite fields is the intersection of the code and its dual, which was introduced by Assmus and Key. In this paper, we develop a method to construct linear codes with trivial hull (LCD codes) and one-dimensional hull by employing the positive characteristic analogues of Gauss sums. These codes are quasi-abelian, and sometimes doubly circulant. Some sufficient conditions for a linear code to be an LCD code (resp. a linear code with one-dimensional hull) are presented. It is worth mentioning that we present a lower bound on the minimum distances of the constructed linear codes. As an application, using these conditions, we obtain some optimal or almost optimal LCD codes (resp. linear codes with one-dimensional hull) with respect to the online Database of Grassl.
In this work we study evaluation codes defined on the points of a subset X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepack...
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In this work we study evaluation codes defined on the points of a subset X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {X}$$\end{document} of an affine space over a finite field, whose vanishing ideal admits a Grobner basis of a certain type, which occurs for subsets considered in several well-known examples of evaluation codes, like Reed-Solomon codes, Reed-Muller codes and affine cartesian codes. We determine properties of the polynomials in this basis which allow the determination of the footprint of the vanishing ideal and the explicit construction of indicator functions for the points of X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {X}$$\end{document}. We then consider generalized monomial evaluation codes and find information on their duals, and the dimension of their hulls. We present several examples of applications of the results we found.
We present two new constructions of entanglement-assisted quantum error-correcting codes using some fundamental properties of (classical) linear codes in an effective way. The main ideas include linear complementary d...
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We present two new constructions of entanglement-assisted quantum error-correcting codes using some fundamental properties of (classical) linear codes in an effective way. The main ideas include linear complementary dual codes and related concatenation constructions. Numerical examples in modest lengths show that our constructions perform better than known constructions in the literature. We also give a proof on a generalization of binary Singleton type bound on entanglement-assisted quantum error-correcting codes to arbitrary q-ary entanglement-assisted quantum error-correcting codes.
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