The Weight Enumerator of Three Families of Cyclic Codes

Cyclic codes are a subclass of linear codes and have wide applications in consumer electronics, data storage systems, and communication systems due to their efficient encoding and decoding algorithms. Cyclic codes with many zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a very small number of cases. In general the calculation of the weight distribution of cyclic codes is heavily based on the evaluation of some exponential sums over finite fields. Very recently, Li, Hu, Feng and Ge studied a class of $p$-ary cyclic codes of length $p^{2m}-1$, where $p$ is a prime and $m$ is odd. They determined the weight distribution of this class of cyclic codes by establishing a connection between the involved exponential sums with the spectrum of Hermitian forms graphs. In this paper, this class of $p$-ary cyclic codes is generalized and the weight distribution of the generalized cyclic codes is settled for both even $m$ and odd $m$ alone with the idea of Li, Hu, Feng, and Ge. The weight distributions of two related families of cyclic codes are also determined.Comment: 13 Pages, 3 Table

Crooked maps in F2n

AbstractA map f:F2n→F2n is called crooked if the set {f(x+a)+f(x):x∈F2n} is an affine hyperplane for every fixed a∈F2n∗ (where F2n is considered as a vector space over F2). We prove that the only crooked power maps are the quadratic maps x2i+2j with gcd(n,i−j)=1. This is a consequence of the following result of independent interest: for any prime p and almost all exponents 0⩽d⩽pn−2 the set {xd+γ(x+a)d:x∈Fpn} contains n linearly independent elements, where γ and a≠0 are arbitrary elements from Fpn

Weight distribution of cyclic codes defined by quadratic forms and related curves

We consider cyclic codes CL associated to quadratic trace forms inm variables (Formula Presented) determined by a family L of q-linearized polynomials R over Fqm, and three related codes CL,0, CL,1, and CL,2. We describe the spectra for all these codes when L is an even rank family, in terms of the distribution of ranks of the forms QR in the family L, and we also computethe complete weight enumerator for CL. In particular, considering the family L = ‹xql›, with l fixed in N, we give the weight distribution of four parametrized families of cyclic codes Cl, Cl,0,Cl,1, and Cl,2 over Fq with zeros(Formula Presented) respectively,where q = ps with p prime, α is a generator of F*qm, and m/(m,l)is even. Finally, we give simple necessary and sufficient conditions for Artin–Schreier curves yp−y = xR(x)+βx, p prime, associated to polynomials R ∈ L to be optimal. We then obtain several maximal and minimal such curves inthe case (Formula Presented).Fil: Podesta, Ricardo Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; ArgentinaFil: Videla Guzman, Denis Eduardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentin

The spectra of generalized Paley graphs of qℓ+1q^\ell+1 powers and applications

We consider a special class of generalized Paley graphs over finite fields, namely the Cayley graphs with vertex set $\mathbb{F}_{q^m}$ and connection set the nonzero $(q^\ell+1)$-th powers in $\mathbb{F}_{q^m}$, as well as their complements. We explicitly compute the spectrum of these graphs. As a consequence, the graphs turn out to be (with trivial exceptions) simple, connected, non-bipartite, integral and strongly regular (of Latin square type in half of the cases). By using the spectral information we compute several invariants of these graphs. We exhibit infinite families of pairs of equienergetic non-isospectral graphs. As applications, on the one hand we solve Waring's problem over $\mathbb{F}_q^m$ for the exponents $q^\ell+1$, for each $q$ and for infinite values of $\ell$ and $m$. We obtain that the Waring's number $g(q^\ell+1,q^m)=1$ or $2$, depending on $m$ and $\ell$, thus tackling some open cases. On the other hand, we construct infinite towers of Ramanujan graphs in all characteristics.Comment: 27 pages, 3 tables. A little modification of the title. Corollary 4.8 removed. Added Section 6 on "Energy". Minor typos corrected. Ihara zeta functions at the end correcte

Cross-Correlations of Geometric Sequences in Characteristic Two

Cross-correlation functions are determined for a large class of geometric sequences based on m-sequences in characteristic two. These sequences are shown to have low cross-correlation values in certain cases. They are also shown to have signi cantly higher linear complexities than previously studied geometric sequences. These results show that geometric sequences are candidates for use in spread-spectrum communications systems in which cryptographic security is a factor