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We present an efficient list decoding algorithm in the style of Guruswami-Sudan for algebraic geometry codes. Our decoder can decode any such code using $\tilde{\mathcal {O}} (s\ell ^{\omega }\mu ^{\omega -1}(n+g))$ operations in the underlying finite field, where $n$ is the code length, $g$ is the genus of the function field used to construct the code, $s$ is the multiplicity parameter, $\ell $ is the designed list size and $\mu $ is the smallest positive element in the Weierstrass semigroup at some chosen place; the “soft-O” notation $\tilde{\mathcal {O}} (\cdot)$ is similar to the “big-O” notation $\mathcal {O}(\cdot)$ , but ignores logarithmic factors. For the interpolation step, which constitutes the computational bottleneck of our approach, we use known algorithms for univariate polynomial matrices, while the root-finding step is solved using existing algorithms for root-finding over univariate power series. [ABSTRACT FROM AUTHOR]

In 1964, Massey introduced a class of codes with complementary duals which are called linear complimentary dual (LCD) codes. He showed that LCD codes have applications in communication system, side-channel attack and so on. LCD codes have been extensively studied in literature. On the other hand, MDS codes form an optimal family of classical codes which have wide applications in both theory and practice. The main purpose of this paper is to give an explicit construction of several classes of LCD MDS codes, using tools from algebraic function fields. We exemplify this construction and obtain several classes of explicit LCD MDS codes for the odd characteristic case. [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]

Abstract: In this article we study Drinfeld modular curves associated to congruence subgroups of where p is a prime of . For we compute the extension degrees and investigate the structure of the Galois closures of the covers and some of their variations. The results have some immediate implications for the Galois closures of two well-known optimal wild towers of function fields over finite fields introduced by Garcia and Stichtenoth, for which the modular interpretation was given by Elkies. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we derive “normal forms” for the defining equations of recursive towers of function fields over finite fields, under certain weak hypotheses. Specially interesting are the cases of towers of Kummer type and towers of Artin–Schreier type. [Copyright &y& Elsevier]

The Gilbert–Varshamov bound guarantees the existence of families of codes over the finite field \BBF\ell with good asymptotic parameters. We show that this bound can be improved for all nonprime fields \BBF\ell with $\ell\geq 49$ , except possibly $\ell=125$ . We observe that the same improvement even holds within the class of transitive codes and within the class of self-orthogonal codes. [ABSTRACT FROM AUTHOR]

The second generalized GK function fields K n are a recently found family of maximal function fields over the finite field with q 2 n elements, where q is a prime power and n ≥ 1 an odd integer. In this paper we construct many new maximal function fields by determining various Galois subfields of K n. In case gcd ⁡ (q + 1 , n) = 1 and either q is even or q ≡ 1 (mod 4) , we find a complete list of Galois subfields of K n. Our construction adds several previously unknown genera to the genus spectrum of maximal curves. [ABSTRACT FROM AUTHOR]