American Institute of Mathematical Sciences

Two skew cyclic codes can be equivalent for the Hamming metric only if they have the same length, and only the zero code is degenerate. The situation is completely different for the rank metric. We study rank equivalences between skew cyclic codes of different lengths and, with the aim of finding the skew cyclic code of smallest length that is rank equivalent to a given one, we define different types of length for a given skew cyclic code, relate them and compute them in most cases. We give different characterizations of rank degenerate skew cyclic codes using conventional polynomials and linearized polynomials. Some known results on the rank weight hierarchy of cyclic codes for some lengths are obtained as particular cases and extended to all lengths and to all skew cyclic codes. Finally, we prove that the smallest length of a linear code that is rank equivalent to a given skew cyclic code can be attained by a pseudo-skew cyclic code.

In previous work [3] Colón-Reyes et al developed criteria for determining when a discrete monomial dynamical system reaches steady state behaviour. These criteria depend on determining when a certain matrix over a finite ring, that is not a field, defines a fixed point system. It was not until recently that criteria to determine linear steady state behaviour over rings have been found. Using these new results we present a new algorithm to determine steady state behaviour of monomial dynamical systems over finite fields. Delgado-Eckert [5] has also obtained an algorithm for the finite field case, but his algorithm does not take into account the result in [3] and requires $O(n^4\; q^2 \log\; q)$ integer operations. Our algorithm requires only $O(n^3 \log(n\; \log \; q))$ integer operations.

We construct a family of sextics from the Wiman and Edge sextics. We find a curve over $\mathbb{F}_{5^7}$ attaining the Serre bound, and update $9$ entries of genus $6$ in manYPoints.org by computer search on these sextics.

We investigate explicit constructions of bent functions which are linear on elements of spreads. Our constructions are obtained from symplectic presemifields which are associated to pseudo-planar functions. The following diagram gives an indication of the main interconnections arising in this paper: \begin{document} $pseudo-planar\ functions \longleftrightarrow\ commutaive\ presemifields \longrightarrow bent\ functions$ \end{document}

Permutations of finite fields have important applications in cryptography and coding theory. Involutions are permutations that are its own inverse and are of particular interest because the implementation used for coding can also be used for decoding. We present explicit formulas for all the involutions of \begin{document} ${\mathbb{ F\!}}_q$ \end{document} that are given by monomials and for their fixed points.

In this paper we study the structure of specific linear codes called DNA codes. The first attempts on studying such codes have been proposed over four element rings which are naturally matched with DNA four letters. Later, double (pair) DNA strings or more general \begin{document} $k$ \end{document}-DNA strings called \begin{document} $k$ \end{document}-mers have been matched with some special rings and codes over such rings with specific properties are studied. However, these matchings in general are not straightforward and because of the fact that the reverse of the codewords (\begin{document} $k$ \end{document}-mers) need to exist in the code, the matching problem is difficult and it is referred to as the reversibility problem. Here, \begin{document} $8$ \end{document}-mers (DNA 8-bases) are matched with the ring elements of \begin{document} $R_{16}=F_{16}+uF_{16}+vF_{16}+uvF_{16}.$ \end{document} Furthermore, cyclic codes over the ring \begin{document} $R_{16}$ \end{document} where the multiplication is taken to be noncommutative with respect to an automorphism \begin{document} $\theta$ \end{document} are studied. The preference on the skewness is shown to be very useful and practical especially since this serves as a direct solution to the reversibility problem compared to the commutative approaches.

We present a 3D array construction with application to video watermarking. This new construction uses two main ingredients: an extended rational cycle (ERC) as a shift sequence and a Legendre array as a base. This produces a family of 3D arrays with good auto and cross-correlation. We calculate exactly the values of the auto correlation and the cross-correlation function and their frequency. We present a unified method of obtaining multivariate recursion polynomials and their footprints for all finite multidimensional arrays. Also, we describe new results for arbitrary arrays and enunciate a result for arrays constructed using the method of composition. We also show that the size of the footprint is invariant under dimensional transformations based on the Chinese Remainder Theorem.

We compute the covering radius of some families of binary cyclic codes. In particular, we compute the covering radius of cyclic codes with two zeros and minimum distance greater than 3. We also compute the covering radius of some binary primitive BCH codes over \begin{document}$\mathbb{F}_{2^f}$\end{document}, where \begin{document}$f=7, 8$\end{document}.

The paper presents methods for designing functions having many applications in particular to construct linear codes with few weights. The former codes have several applications in secret sharing, authentication codes, association schemes and strongly regular graphs. We firstly provide new secondary constructions of bent functions generalizing the well-known Rothaus' constructions as well as their dual functions. From our generalization, we show that we are able to compute the dual function of a bent function built from Rothaus' construction. Next we present a result leading to a new method for constructing semi-bent functions and few Walsh transform values functions built from bent functions.

In this paper, we firstly compute the dual functions of elementary symmetric bent functions. Next, we derive a new secondary construction of bent functions (given with their dual functions) involving symmetric bent functions, leading to a generalization of the well-know Rothaus' construction.

A code is said to be complementary dual if it meets its dual trivially. We give a sufficient condition for a special class of additive cyclic codes to be complementary dual.

We consider two metrics decoding equivalent if they impose the same minimum distance decoding for every code. It is known that, up to this equivalence, every metric is isometrically embeddable into the Hamming cube.

We present an algorithm which for any translation invariant metric gives an upper bound on the minimum dimension of such an embedding. We also give lower and upper bounds for this embedding dimension over the set of all such metrics.

We present an N-dimensional generalisation of the two-dimensional block-circulant perfect array construction by Blake et al. As in Blake et al, the families of N-dimensional arrays possess pairwise good zero correlation zone (ZCZ) cross-correlation. Both constructions use a perfect autocorrelation sequence with the array orthogonality property (AOP).

Since 1970, Boolean functions have been the focus of a lot of attention in cryptography. An important topic in symmetric ciphers concerns the cryptographic properties of Boolean functions and constructions of Boolean functions with good cryptographic properties, that is, good resistance to known attacks. An important progress in cryptanalysis areas made in 2003 was the introduction by Courtois and Meier of algebraic attacks and fast algebraic attacks which are very powerful analysis concepts and can be applied to almost all cryptographic algorithms. To study the resistance against algebraic attacks, the notion of algebraic immunity has been introduced. In this paper, we use a parameter introduced by Liu and al., called fast algebraic immunity, as a tool to measure the resistance of a cryptosystem (involving Boolean functions) to fast algebraic attacks. We prove an upper bound on the fast algebraic immunity. Using our upper bound, we establish the weakness of trace inverse functions against fast algebraic attacks confirming a recent result of Feng and Gong.

In this article we describe how to find the parameters of subfield subcodes of extended Norm-Trace codes (ENT codes). With a Gröbner basis of the ideal of the \begin{document}$\mathbb{F}_{q^r}$\end{document} rational points of the extended Norm-Trace curve one can determine the dimension of the subfield subcodes or the dimension of the trace code. We also find a BCH-like bound from the minimum distance of the original code. The ENT codes we study here are a more general class of codes than those given in [1]. We study their subfield subcodes as well. We give an example of ENT subfield subcodes that have optimal parameters. Furthermore, we give examples of binary subfield subcodes of ENT codes of very large length for modern applications (e.g. for flash memories).

An almost perfect nonlinear (APN) function \begin{document}$f:\mathbb{F}_{2^n}\rightarrow\mathbb{F}_{2^n}$\end{document} (necessarily polynomial) is called exceptional APN if it is APN on infinitely many extensions of \begin{document}$\mathbb{F}_{2^n}$\end{document}. Aubry, McGuire and Rodier conjectured that the only exceptional APN functions are the Gold and the Kasami-Welch monomial functions. They established that a polynomial function of odd degree is not exceptional APN provided the degree is not a Gold number \begin{document}$(2^k+1)$\end{document} or a Kasami-Welch number \begin{document}$(2^{2k}-2^k+1)$\end{document}. When the degree of the polynomial function is a Gold number or a Kasami-Welch number, several partial results have been obtained by several authors including us. In this article we address these exceptions. We almost prove the exceptional APN conjecture in the Gold degree case when \begin{document}$\deg{(h(x))}$\end{document} is odd. We also show exactly when the corresponding multivariate polynomial \begin{document}$φ(x, y, z)$\end{document} is absolutely irreducible. Also, there is only one result known when \begin{document}$f(x)=x^{2^{k}+1} + h(x)$\end{document}, and \begin{document}$\deg(h(x))$\end{document} is even. Here, we extend this result as well, thus making progress in this case that seems more difficult.

Vasiga and Shallit [17] study tails and cycles in orbits of iterations of quadratic polynomials over prime fields. These results were extended to repeated exponentiation by Chou and Shparlinski [3]. We show, using the quadratic reciprocity law, that it is possible to extend these results to Rédei functions over prime fields.