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Planar functions are mappings from a finite field F q to itself with an extremal differential property. Such functions give rise to finite projective planes and other combinatorial objects. There is a subtle difference between the definitions of these functions depending on the parity of q and we consider the case that q is even. We classify polynomials of degree at most q 1 / 4 that induce planar functions on F q , by showing that such polynomials are precisely those in which the degree of every monomial is a power of two. As a corollary we obtain a complete classification of exceptional planar polynomials, namely polynomials over F q that induce planar functions on infinitely many extensions of F q. The proof strategy is to study the number of F q -rational points of an algebraic curve attached to a putative planar function. Our methods also give a simple proof of a new partial result for the classification of almost perfect nonlinear functions. [ABSTRACT FROM AUTHOR]

Maximum rank-distance (MRD) codes are extremal codes in the space of m×n matrices over a finite field, equipped with the rank metric. Up to generalizations, the classical examples of such codes were constructed in the 1970s and are today known as Gabidulin codes. Motivated by several recent approaches to construct MRD codes that are inequivalent to Gabidulin codes, we study the equivalence issue for Gabidulin codes themselves. This shows in particular that the family of Gabidulin codes already contains a huge subset of MRD codes that are pairwise inequivalent, provided that 2⩽m⩽n-2. [ABSTRACT FROM AUTHOR]

For integer r satisfying 0\leq r\leq p-2, a sequence family \Omega_r of polyphase sequences of prime period p, size (p-2)p^r, and maximum correlation at most 2+(r+1)\sqrtp is presented. The sequence families are nested, that is, \Omega_r is contained in \Omegar+1, which provides design flexibility with respect to family size and maximum correlation. The sequences in \Omega_r are derived from a combination of multiplicative and additive characters of a prime field. Estimates on hybrid character sums are then used to bound the maximum correlation. This construction generalizes \Omega_0, which was previously proposed by Scholtz and Welch. Sequence family \Omega_2 is closely related to a recent design by Wang and Gong, who bounded its maximum correlation using methods from representation theory and asked for a more direct proof of this bound. Such a proof is given here and an improvement of the bound is provided. [ABSTRACT FROM AUTHOR]

Abstract: Let be the set of symmetric bilinear forms on an m-dimensional vector space over , where q is a power of two. A subset Y of is called an -set if the difference of every two distinct elements in Y has rank at least d. Such objects are closely related to certain families of codes over Galois rings of characteristic four. An upper bound on the size of -sets is derived, and in certain cases, the rank distance distribution of an -set is explicitly given. Constructions of -sets are provided for all possible values of m and d. [ABSTRACT FROM AUTHOR]

In this paper, 74-valued quadratic forms defined on a vector space over GF(2) are studied. A classification of such forms is established, distinguishing Z4-valued quadratic forms only by their rank and whether the associated bilinear form is alternating. This result is used to compute the distribution of certain exponential sums, which occur frequently in the analysis of quaternary codes and quaternary sequence sets. The concept is applied as follows. When t = 0 or m is odd, the correlation distribution of family S (t), consisting of quaternary sequences of length 2m - 1, is established. Then, motivated by practical considerations, a subset S* (t) of family S (t) is defined, and the correlation distribution of family S* (t) is given for odd and even m. [ABSTRACT FROM AUTHOR]

We consider a generalisation of a conjecture by Patterson and Wiedemann from 1983 on the Hamming distance of a function from F q n to F q to the set of affine functions from F q n to F q. We prove the conjecture for each q such that the characteristic of F q lies in a subset of the primes with density 1 and we prove the conjecture for all q by assuming the generalised Riemann hypothesis. Roughly speaking, we show the existence of functions for which the distance to the affine functions is maximised when n tends to infinity. This also determines the asymptotic behaviour of the covering radius of the [ q n , n + 1 ] Reed-Muller code over F q and so answers a question raised by Leducq in 2013. Our results extend the case q = 2 , which was recently proved by the author and which corresponds to the original conjecture by Patterson and Wiedemann. Our proof combines evaluations of Gauss sums in the semiprimitive case, probabilistic arguments, and methods from discrepancy theory. [ABSTRACT FROM AUTHOR]