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Abstract: One of the big questions in the area of curves over finite fields concerns the distribution of the numbers of points: which numbers occur as the number of points on a curve of genus g? The same question can be asked of various subclasses of curves. In this article we classify the possibilities for the number of points on hyperelliptic supersingular curves of genus 4 over finite fields of order , n odd. [Copyright &y& Elsevier]

Abstract: In this article a technique for constructing p-ary bent functions from near-bent functions is presented. This technique is then used to obtain both weakly regular and non-weakly regular bent functions. In particular we present the first known infinite class of non-weakly regular bent functions. [Copyright &y& Elsevier]

Establishing the CCZ-equivalence of a pair of APN functions is generally quite difficult. In some cases, when seeking to show that a putative new infinite family of APN functions is CCZ inequivalent to an already known family, we rely on computer calculation for small values of n. In this paper we present a method to prove the inequivalence of quadratic APN functions with the Gold functions. Our main result is that a quadratic function is CCZ-equivalent to the APN Gold function $${x^{2^r+1}}$$ if and only if it is EA-equivalent to that Gold function. As an application of this result, we prove that a trinomial family of APN functions that exist on finite fields of order 2 where n ≡ 2 mod 4 are CCZ inequivalent to the Gold functions. The proof relies on some knowledge of the automorphism group of a code associated with such a function. [ABSTRACT FROM AUTHOR]

Abstract: We construct Weil numbers corresponding to genus-2 curves with p-rank 1 over the finite field of elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group of -valued points of the Jacobian has prime order, while another allows for a prescribed embedding degree with respect to a subgroup of prescribed order. The curves are defined over out of necessity: we show that curves of p-rank 1 over for large p cannot be efficiently constructed using explicit CM constructions. [Copyright &y& Elsevier]

Abstract: We give a construction of bent functions in dimension 2m from near-bent functions in dimension . In particular, we give the first ever examples of non-weakly-normal bent functions in dimensions 10 and 12, which demonstrates the significance of our construction. [Copyright &y& Elsevier]

Abstract: We present a general criterion for near bent functions to be bent on a hyperplane. Let be odd and let . We show that the Kasami–Welch function is a bent function when restricted to the hyperplane of trace 0 elements in . [Copyright &y& Elsevier]

Abstract: A theorem of Bannai and Sawano shows that certain four-weight spin models exist if and only if certain quasi-3 designs exist. We verify that the known quasi-3 designs, other than SDP designs, do not give rise to four-weight spin models. [Copyright &y& Elsevier]

Abstract: We present an alternative proof of a result of Zeng–Shan–Hu that shows that the cyclic code with three zeros α, , has the same weight distribution as the 3-error-correcting BCH code. Our proof uses the theory of algebraic curves over finite fields, and combines results that are already in the literature. This method is applicable in other cases too. [Copyright &y& Elsevier]

Abstract: Let be a field and let denote the space of matrices over . We investigate properties of a subspace of of dimension in which each non-zero element of has rank at least and enumerate the number of elements of a given rank in when is finite. We also provide an upper bound for the dimension of a constant rank subspace of when is finite and give non-trivial examples to show that our bound is optimal in some cases. We include a similar a bound for the maximum dimension of a constant rank subspace of skew-symmetric matrices over a finite field. [Copyright &y& Elsevier]

Abstract: We study PN and APN functions over the integers modulo . We give some construction techniques based on Costas arrays, which allow us to construct APN permutations on where is a prime. Although PN permutations do not exist, one set of our functions is very close to being a set of PN permutations. [Copyright &y& Elsevier]

Abstract: We introduce two new infinite families of APN functions, one on fields of order for k not divisible by 2, and the other on fields of order for k not divisible by 3. The polynomials in the first family have between three and terms, the second family''s polynomials have three terms. [Copyright &y& Elsevier]