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In 2017 Skabelund constructed two new examples of maximal curves S ˜ q and R ˜ q as covers of the Suzuki and Ree curves, respectively. The resulting Skabelund curves are analogous to the Giulietti–Korchmáros cover of the Hermitian curve. In this paper a complete characterization of all Galois subcovers of the Skabelund curves S ˜ q and R ˜ q is given. Calculating the genera of the corresponding curves, we find new additions to the list of known genera of maximal curves over finite fields. [ABSTRACT FROM AUTHOR]

We consider linear codes over a finite field F q , for odd q , derived from determinantal varieties, obtained from symmetric matrices of bounded ranks. A formula for the weight of a codeword is derived. Using this formula, we have computed the minimum distance for the codes corresponding to matrices upperbounded by any fixed, even rank. A conjecture is proposed for the cases where the upper bound is odd. At the end of the article, tables for the weights of these codes, for spaces of symmetric matrices up to order 5, are given. [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]

In [14] , D. Skabelund constructed a maximal curve over F q 4 as a cyclic cover of the Suzuki curve. In this paper we explicitly determine the structure of the Weierstrass semigroup at any point P of the Skabelund curve. We show that its Weierstrass points are precisely the F q 4 -rational points. Also we show that among the Weierstrass points, two types of Weierstrass semigroup occur: one for the F q -rational points, one for the remaining F q 4 -rational points. For each of these two types its Apéry set is computed as well as a set of generators. [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]