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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 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 this article, we present a new construction of evaluation codes in the Hamming metric, which we call twisted Reed–Solomon codes. Whereas Reed–Solomon (RS) codes are MDS codes, this need not be the case for twisted RS codes. Nonetheless, we show that our construction yields several families of MDS codes. Further, for a large subclass of (MDS) twisted RS codes, we show that the new codes are not generalized RS codes. To achieve this, we use properties of Schur squares of codes as well as an explicit description of the dual of a large subclass of our codes. We conclude the paper with a description of a decoder, that performs very well in practice as shown by extensive simulation results. [ABSTRACT FROM AUTHOR]

In this article we prove a conjecture formulated by A. B. Sørensen in 1991 on the maximal number of Fq2-rational points on the intersection of a non-degenerate Hermitian surface and a surface of degree d ≤ q. [ABSTRACT FROM AUTHOR]

We investigate algorithms for encoding of one-point algebraic geometry (AG) codes over certain plane curves called $C_{ab}$ curves, as well as algorithms for inverting the encoding map, which we call “unencoding”. Some $C_{ab}$ curves have many points or are even maximal, e.g. the Hermitian curve. Our encoding resp. unencoding algorithms have complexity $ \tilde { \mathcal {O}}(\text {n}^{3/2})$ resp. $ \tilde { \mathcal {O}}({\it\text { qn}})$ for AG codes over any $C_{ab}$ curve satisfying very mild assumptions, where n is the code length and q the base field size, and $ \tilde { \mathcal {O}}$ ignores constants and logarithmic factors in the estimate. For codes over curves whose evaluation points lie on a grid-like structure, for example the Hermitian curve and norm-trace curves, we show that our algorithms have quasi-linear time complexity $ \tilde { \mathcal {O}}(\text {n})$ for both operations. For infinite families of curves whose number of points is a constant factor away from the Hasse-Weil bound, our encoding and unencoding algorithms have complexities $ \tilde { \mathcal {O}}(\text {n}^{5/4})$ and $ \tilde { \mathcal {O}}(\text {n}^{3/2})$ respectively. [ABSTRACT FROM AUTHOR]

In this article we explicitly determine the Weierstrass semigroup at any point and the full automorphism group of a known F q 2 -maximal curve X 3 having the third largest genus. This curve arises as a Galois subcover of the Hermitian curve, and its uniqueness (with respect to the value of its genus) is a well-known open problem. Knowing the Weierstrass semigroups may provide a key towards solving this problem. Surprisingly enough X 3 has many different types of Weierstrass semigroups and the set of its Weierstrass points is much richer than its set of F q 2 -rational points. This makes the curve X 3 the first explicitly known maximal curve having non-rational Weierstrass points. We show that a similar exceptional behaviour does not occur in terms of automorphisms, that is, Aut (X 3) is exactly the automorphism group inherited from the Hermitian curve, apart from small values of q. [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]

In this article we consider linear codes coming from skew-symmetric determinantal varieties, which are defined by the vanishing of minors of a certain fixed size in the space of skew-symmetric matrices. In odd characteristic, the minimum distances of these codes are determined and a recursive formula for the weight of a general codeword in these codes is given. [ABSTRACT FROM AUTHOR]

In this note, we give a very simple description of the generalized Hamming weights of Reed–Muller codes. For this purpose, we generalize the well-known Macaulay representation of a nonnegative integer and state some of its basic properties. [ABSTRACT FROM AUTHOR]

We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Gröbner basis of the ideal. Further we give a projective analogue for the so-called footprint bound, and a version of it that is suitable for estimating the number of rational points of projective algebraic varieties over finite fields. An application to Serre's inequality for the number of points of projective hypersurfaces over finite fields is included. [ABSTRACT FROM AUTHOR]

In this article we construct for any prime power q and odd n⩾5, a new Fq2n‐maximal curve Xn. Like the Garcia–Güneri–Stichtenoth maximal curves, our curves generalize the Giulietti–Korchmáros maximal curve, though in a different way. We compute the full automorphism group of Xn, yielding that it has precisely q(q2−1)(qn+1) automorphisms. Further, we show that unless q=2, the curve Xn is not a Galois subcover of the Hermitian curve. Finally, up to our knowledge, we find new values of the genus spectrum of Fq2n‐maximal curves, by considering some Galois subcovers of Xn. [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]

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]

We improve previously known lower bounds for the minimum distance of certain two-point AG codes constructed using a Generalized Giulietti–Korchmaros curve (GGK). Castellanos and Tizziotti recently described such bounds for two-point codes coming from the Giulietti–Korchmaros curve. Our results completely cover and in many cases improve on their results, using different techniques, while also supporting any GGK curve. Our method builds on the order bound for AG codes: to enable this, we study certain Weierstrass semigroups. This allows an efficient algorithm for computing our improved bounds. We find several new improvements upon the MinT minimum distance tables. [ABSTRACT FROM AUTHOR]

In this article we explicitly determine the structure of the Weierstrass semigroups H ( P ) for any point P of the Giulietti–Korchmáros curve X . We show that as the point varies, exactly three possibilities arise: one for the F q 2 -rational points (already known in the literature), one for the F q 6 ∖ F q 2 -rational points, and one for all remaining points. As a result, we prove a conjecture concerning the structure of H ( P ) in case P is an F q 6 ∖ F q 2 -rational point. [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]

About two decades ago, Tsfasman and Boguslavsky conjectured a formula for the maximum number of common zeros that r linearly independent homogeneous polynomials of degree d in m + 1 variables with coefficients in a finite field with q elements can have in the corresponding m-dimensional projective space. Recently, it has been shown by Datta and Ghorpade that this conjecture is valid if r is at most m + 1 and can be invalid otherwise. Moreover a new conjecture was proposed for many values of r beyond m + 1. In this paper, we prove that this new conjecture holds true for several values of r. In particular, this settles the new conjecture completely when d = 3. Our result also includes the positive result of Datta and Ghorpade as a special case. Further, we determine the maximum number of zeros in certain cases not covered by the earlier conjectures and results, namely, the case of d = q−1 and of d = q. All these results are directly applicable to the determination of the maximum number of points on sections of Veronese varieties by linear subvarieties of a fixed dimension, and also the determination of generalized Hamming weights of projective Reed-Muller codes. [ABSTRACT FROM AUTHOR]

We consider a class of linear codes associated to projective algebraic varieties defined by the vanishing of minors of a fixed size of a generic matrix. It is seen that the resulting code has only a small number of distinct weights. The case of varieties defined by the vanishing of 2×2 minors is considered in some detail. Here we obtain the complete weight distribution. Moreover, several generalized Hamming weights are determined explicitly and it is shown that the first few of them coincide with the distinct nonzero weights. One of the tools used is to determine the maximum possible number of matrices of rank 1 in a linear space of matrices of a given dimension over a finite field. In particular, we determine the structure and the maximum possible dimension of linear spaces of matrices in which every nonzero matrix has rank 1. [ABSTRACT FROM AUTHOR]

We present the first two sub-quadratic complexity decoding algorithms for one-point Hermitian codes. The first is based on a fast realization of the Guruswami–Sudan algorithm using state-of-the-art algorithms from computer algebra for polynomial-ring matrix minimization. The second is a power decoding algorithm: an extension of classical key equation decoding which gives a probabilistic decoding algorithm up to the Sudan radius. We show how the resulting key equations can be solved by the matrix minimization algorithms from computer algebra, yielding similar asymptotic complexities. [ABSTRACT FROM AUTHOR]

We derive the Wu list-decoding algorithm for generalized Reed–Solomon (GRS) codes by using Gröbner bases over modules and the Euclidean algorithm as the initial algorithm instead of the Berlekamp–Massey algorithm. We present a novel method for constructing the interpolation polynomial fast. We give a new application of the Wu list decoder by decoding irreducible binary Goppa codes up to the binary Johnson radius. Finally, we point out a connection between the governing equations of the Wu algorithm and the Guruswami–Sudan algorithm, immediately leading to equality in the decoding range and a duality in the choice of parameters needed for decoding, both in the case of GRS codes and in the case of Goppa codes. [ABSTRACT FROM PUBLISHER]

Affine Grassmann codes are a variant of generalized Reed–Muller codes and are closely related to Grassmann codes. These codes were introduced in a recent work by Beelen Here, we consider, more generally, affine Grassmann codes of a given level. We explicitly determine the dual of an affine Grassmann code of any level and compute its minimum distance. Further, we ameliorate the results by Beelen concerning the automorphism group of affine Grassmann codes. Finally, we prove that affine Grassmann codes and their duals have the property that they are linear codes generated by their minimum-weight codewords. This provides a clean analogue of a corresponding result for generalized Reed–Muller codes. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper we prove a conjecture by Schweizer on the reduction of the Drinfeld modular polynomial modulo . The proof mainly involves manipulations of binomial coefficients in characteristic p. [Copyright &y& Elsevier]

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]

We consider a new class of linear codes, called affine Grassmann codes. These can be viewed as a variant of generalized Reed-Muller codes and are closely related to Grassmann codes. We determine the length, dimension, and the minimum distance of any affine Grassmann code. Moreover, we show that affine Grassmann codes have a large automorphism group and determine the number of minimum weight codewords. [ABSTRACT FROM AUTHOR]

Abstract: A Reed–Solomon code of length can be list decoded using the well-known Guruswami–Sudan algorithm. By a result of the interpolation part in this algorithm can be done in complexity , where denotes the designed list size and the multiplicity parameter. The parameters and are sometimes considered to be constants in the complexity analysis, but for high rate Reed–Solomon codes, their values can be very large. In this paper we will combine ideas from and the concept of key equations to get an algorithm that has complexity . This compares favorably to the complexities of other known interpolation algorithms. [Copyright &y& Elsevier]

Abstract: Baker''s theorem is a theorem giving an upper-bound for the genus of a plane curve. It can be obtained by studying the Newton-polygon of the defining equation of the curve. In this paper we give a different proof of Baker''s theorem not using Newton-polygon theory, but using elementary methods from the theory of function fields (Theorem 2.4). Also we state a generalization to several variables that can be used if a curve is defined by several bivariate polynomials that all have one variable in common (Theorem 3.3). As a side result, we obtain a partial explicit description of certain Riemann–Roch spaces, which is useful for applications in coding theory. We give several examples and compare the bound on the genus we obtain, with the bound obtained from Castelnuovo''s inequality. [Copyright &y& Elsevier]

In this article, we consider the decoding problem of Grassmann codes using majority logic. We show that for two points of the Grassmannian, there exists a canonical geodesic between these points once a complete flag is fixed. These geodesics are used to construct a large set of parity checks orthogonal on a coordinate of the code, resulting in a majority decoding algorithm. [ABSTRACT FROM AUTHOR]

Abstract: The order bound gives an in general very good lower bound for the minimum distance of one-point algebraic geometric codes coming from curves. This paper is about a generalization of the order bound to several-point algebraic geometric codes coming from curves. [Copyright &y& Elsevier]

Abstract: The Weierstrass semigroup is well known and has been studied. Recently there has been a renewed interest in these semigroups because of applications in coding theory. Generalizations of the Weierstrass semigroup to -tuples have been made and studied. We will state and study another possible generalization. [Copyright &y& Elsevier]

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 this paper we state and explore a connection between graph theory and the theory of recursively defined towers. This leads, among other things, to a generalization of Lenstra''s identity (Finite Fields Appl. 8 (2001) 166) and the solution of an open problem concerning the Deuring polynomial posed in (J. Reine Angew. Math. 557 (2003) 53). Further we investigate the effect extension of the constant field has on the limit of certain towers. [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 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]

We determine the number of F q -rational points of hyperplane sections of classical determinantal varieties defined by the vanishing of minors of a fixed size of a generic matrix, and identify the hyperplane sections giving the maximum number of F q -rational points. Further we consider similar questions for sections by linear subvarieties of a fixed codimension in the ambient projective space. This is closely related to the study of linear codes associated to determinantal varieties, and the determination of their weight distribution, minimum distance, and generalized Hamming weights. The previously known results about these are generalized and expanded significantly. Connections to eigenvalues of certain association schemes, distance regular graphs, and rank metric codes are also indicated. [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]

We give a complete characterization of all Galois subfields of the generalized Giulietti–Korchmáros function fields C n / F q 2 n for n ≥ 5 . Calculating the genera of the corresponding fixed fields, we find new additions to the list of known genera of maximal function fields. [ABSTRACT FROM AUTHOR]

In this article we give a Drinfeld modular interpretation for various towers of function fields meeting Zink's bound. [ABSTRACT FROM AUTHOR]

Background The developmental trajectory of psychopathy seemingly begins early in life and includes the presence of callous-unemotional (CU) traits (e.g., deficient emotional reactivity, callousness) in conduct-disordered (CD) youth. Though subregion-specific anomalies in amygdala function have been suggested in CU pathophysiology among antisocial populations, system-level studies of CU traits have typically examined the amygdala as a unitary structure. Hence, nothing is yet known of how amygdala subregional network function may contribute to callous-unemotionality in severely antisocial people. Methods We addressed this important issue by uniquely examining the intrinsic functional connectivity of basolateral amygdala (BLA) and centromedial amygdala (CMA) networks across three matched groups of juveniles: CD offenders with CU traits (CD/CU+; n = 25), CD offenders without CU traits (CD/CU−; n = 25), and healthy control subjects ( n = 24). We additionally examined whether perturbed amygdala subregional connectivity coincides with altered volume and shape of the amygdaloid complex. Results Relative to CD/CU− and healthy control youths, CD/CU+ youths showed abnormally increased BLA connectivity with a cluster that included both dorsal and ventral portions of the anterior cingulate and medial prefrontal cortices, along with posterior cingulate, sensory associative, and striatal regions. In contrast, compared with CD/CU− and healthy control youths, CD/CU+ youths showed diminished CMA connectivity with ventromedial/orbitofrontal regions. Critically, these connectivity changes coincided with local hypotrophy of BLA and CMA subregions (without being statistically correlated) and were associated to more severe CU symptoms. Conclusions These findings provide unique insights into a putative mechanism for perturbed attention-emotion interactions, which could bias salience processing and associative learning in youth with CD/CU+. [ABSTRACT FROM AUTHOR]