منابع مشابه
On Representations of Algebraic-Geometric Codes
We show that all algebraic-geometric codes possess a succinct representation that allows for the list decoding algorithms of [9, 6] to run in polynomial time. We do this by presenting a root-finding algorithm for univariate polynomials over function fields when their coefficients lie in finite-dimensional linear spaces, and proving that there is a polynomial size representation given which the ...
متن کاملLinear representations of convolutional codes over rings
In this paper we extend the relation between convolutional codes and linear systems over finite fields to certain commutative rings through first order representations . We introduce the definition of rings with representations as those for which these representations always exist, and we show that finite products of finite fields belong to this class. We develop the input/state/output represen...
متن کاملWhich linear codes are algebraic-geometric?
An infinite series of curves is constructed in order to show that all linear codes can be obtained from curves using Goppa's construction. If one imposes conditions on the degree of the divisor used, then we derive criteria for linear codes to be algebraic-geometric. In particular, we investigate the family of q-ary Hamming codes, and prove that only those with redundancy one or two, and the bi...
متن کاملNew Ring-Linear Codes from Geometric Dualization
In the 1960s and 1970s the Nordstrom-Robinson-Code [30] and subsequently the infinite series of the Preparata[31], Kerdock[21], Delsarte-Goethals[6] and Goethals-Codes [7] were discovered. Apart from a few corner cases, all of these codes are non-linear binary block codes that have higher minimum distance than any known comparable (having equal size and length) linear binary code. We will call ...
متن کاملLinear Secret Sharing from Algebraic-Geometric Codes
It is well-known that the linear secret-sharing scheme (LSSS) can be constructed from linear error-correcting codes (Brickell [1], R.J. McEliece and D.V.Sarwate [2],Cramer, el.,[3]). The theory of linear codes from algebraic-geometric curves (algebraic-geometric (AG) codes or geometric Goppa code) has been well-developed since the work of V.Goppa and Tsfasman, Vladut, and Zink( see [17], [18] a...
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ژورنال
عنوان ژورنال: Advances in Mathematics
سال: 2015
ISSN: 0001-8708
DOI: 10.1016/j.aim.2015.06.011