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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]

Let X=X(n,q) be the set of n×n Hermitian matrices over Fq2. It is well known that X gives rise to a metric translation association scheme whose classes are induced by the rank metric. We study d-codes in this scheme, namely subsets Y of X with the property that, for all distinct A,B∈Y, the rank of A-B is at least d. We prove bounds on the size of a d-code and show that, under certain conditions, the inner distribution of a d-code is determined by its parameters. Except if n and d are both even and 4≤d≤n-2, constructions of d-codes are given, which are optimal among the d-codes that are subgroups of (X,+). This work complements results previously obtained for several other types of matrices over finite fields. [ABSTRACT FROM AUTHOR]

A Barker sequence is a binary sequence for which all nontrivial aperiodic autocorrelations are at most 1 in magnitude. An old conjecture due to Turyn asserts that there is no Barker sequence of length greater than 13. In 1961, Turyn and Storer gave an elementary, though somewhat complicated, proof that this conjecture holds for odd lengths. We give a new and simpler proof of this result. [ABSTRACT FROM AUTHOR]

Planar functions are special functions from a finite field to itself that give rise to finite projective planes and other combinatorial objects. We consider polynomials over a finite field $$K$$ that induce planar functions on infinitely many extensions of $$K$$ ; we call such polynomials exceptional planar. Exceptional planar monomials have been recently classified. In this paper we establish a partial classification of exceptional planar polynomials. This includes results for the classical planar functions on finite fields of odd characteristic and for the recently proposed planar functions on finite fields of characteristic two. [ABSTRACT FROM AUTHOR]

The extent to which a sequence of finite length differs from a shifted version of itself is measured by its aperiodic autocorrelations. Of particular interest are sequences whose entries are 1 or $$-1$$ , called binary sequences, and sequences whose entries are complex numbers of unit magnitude, called unimodular sequences. Since the 1950s, there is sustained interest in sequences with small aperiodic autocorrelations relative to the sequence length. One of the main motivations is that a sequence with small aperiodic autocorrelations is intrinsically suited for the separation of signals from noise, and therefore has natural applications in digital communications. This survey reviews the state of knowledge concerning the two central problems in this area: How small can the aperiodic autocorrelations of a binary or a unimodular sequence collectively be and how can we efficiently find the best such sequences? Since the analysis and construction of sequences with small aperiodic autocorrelations is closely tied to the (often much easier) analysis of periodic autocorrelation properties, several fundamental results on corresponding problems in the periodic setting are also reviewed. [ABSTRACT FROM AUTHOR]

Let f be a function from $$\mathbb {Z}_q^m$$ to $$\mathbb {Z}_q$$ . Such a function f is bent if all values of its Fourier transform have absolute value 1. Bent functions are known to exist for all pairs $$(m,q)$$ except when m is odd and $$q\equiv 2\pmod 4$$ and there is overwhelming evidence that no bent function exists in the latter case. In this paper the following problem is studied: how closely can the largest absolute value of the Fourier transform of f approach 1? For $$q=2$$ , this problem is equivalent to the old and difficult open problem of determining the covering radius of the first order Reed-Muller code. The main result is, loosely speaking, that the largest absolute value of the Fourier transform of f can be made arbitrarily close to 1 for q large enough. [ABSTRACT FROM AUTHOR]

For odd integer m ≥ 3 and $${t=0,1,\ldots,\frac{m-1}{2}}$$, we define Family $${\fancyscript{V}(t)}$$ to be a set of size 2 containing binary sequences of period 2 − 2. The nontrivial correlations between sequences in Family $${\fancyscript{V}(t)}$$ are bounded in magnitude by 2 + 2. Families $${\fancyscript{V}(0)}$$ and $${\fancyscript{V}(1)}$$ compare favourably to the small and large Kasami sets, respectively. So far, the correlation distribution of Family $${\fancyscript{V}(t)}$$ is only known for t = 0. A general framework for computing the correlation distribution of Family $${\fancyscript{V}(t)}$$ is established. The correlation distribution of $${\fancyscript{V}(1)}$$ is derived, and a way to obtain the correlation distribution of $${\fancyscript{V}(2)}$$ is described. [ABSTRACT FROM AUTHOR]