On a bound of Garcia and Voloch for the number of points of a Fermat curve over a prime field

In 1988 Garcia and Voloch proved the upper bound 4n^{4/3}(p-1)^{2/3} for the number of solutions over a prime finite field F_p of the Fermat equation x^n+y^n=a, where a \in F_p^* and n \ge 2 is a divisor of p-1 such that (n-1/2)^4 \ge p-1. This is better than Weil's bound p+1+(n-1)(n-2)p^{1/2} in the stated range. By refining Garcia and Voloch's proof we show that the constant 4 in their bound can be replaced by 3\cdot 2^{-2/3}.Comment: 4 page

Spectra of phase point operators in odd prime dimensions and the extended Clifford group

We analyse the role of the Extended Clifford group in classifying the spectra of phase point operators within the framework laid out by Gibbons et al for setting up Wigner distributions on discrete phase spaces based on finite fields. To do so we regard the set of all the discrete phase spaces as a symplectic vector space over the finite field. Auxiliary results include a derivation of the conjugacy classes of ${\rm ESL}(2, \mathbb{F}_N)$.Comment: Latex, 19page

Permutation polynomials on matrices

AbstractFamilies of examples are presented of polynomials over a finite field or a residue class ring of the integers, which, on substitution, permute the n×n matrices over that field or residue class ring

A Simple Computational Model for Acceptance/Rejection of Binary Sequence Generators

A simple binary model to compute the degree of balancedness in the output sequence of LFSR-combinational generators has been developed. The computational method is based exclusively on the handling of binary strings by means of logic operations. The proposed model can serve as a deterministic alternative to existing probabilistic methods for checking balancedness in binary sequence generators. The procedure here described can be devised as a first selective criterium for acceptance/rejection of this type of generators.Comment: 16 pages, 0 figure

Efficient Quantum Circuits for Non-Qubit Quantum Error-Correcting Codes

We present two methods for the construction of quantum circuits for quantum error-correcting codes (QECC). The underlying quantum systems are tensor products of subsystems (qudits) of equal dimension which is a prime power. For a QECC encoding k qudits into n qudits, the resulting quantum circuit has O(n(n-k)) gates. The running time of the classical algorithm to compute the quantum circuit is O(n(n-k)^2).Comment: 18 pages, submitted to special issue of IJFC

Pure Asymmetric Quantum MDS Codes from CSS Construction: A Complete Characterization

Using the Calderbank-Shor-Steane (CSS) construction, pure $q$-ary asymmetric quantum error-correcting codes attaining the quantum Singleton bound are constructed. Such codes are called pure CSS asymmetric quantum maximum distance separable (AQMDS) codes. Assuming the validity of the classical MDS Conjecture, pure CSS AQMDS codes of all possible parameters are accounted for.Comment: Change in authors' list. Accepted for publication in Int. Journal of Quantum Informatio

On Mathon's construction of maximal arcs in Desarguesian planes. II

In a recent paper [M], Mathon gives a new construction of maximal arcs which generalizes the construction of Denniston. In relation to this construction, Mathon asks the question of determining the largest degree of a non-Denniston maximal arc arising from his new construction. In this paper, we give a nearly complete answer to this problem. Specifically, we prove that when $m\geq 5$ and $m\neq 9$, the largest $d$ of a non-Denniston maximal arc of degree $2^d$ in PG(2,2^m) generated by a {p,1}-map is (\floor {m/2} +1). This confirms our conjecture in [FLX]. For {p,q}-maps, we prove that if $m\geq 7$ and $m\neq 9$, then the largest $d$ of a non-Denniston maximal arc of degree $2^d$ in PG(2,2^m) generated by a {p,q}-map is either \floor {m/2} +1 or \floor{m/2} +2.Comment: 21 page

A Complete Characterization of Irreducible Cyclic Orbit Codes and their Pl\"ucker Embedding

Constant dimension codes are subsets of the finite Grassmann variety. The study of these codes is a central topic in random linear network coding theory. Orbit codes represent a subclass of constant dimension codes. They are defined as orbits of a subgroup of the general linear group on the Grassmannian. This paper gives a complete characterization of orbit codes that are generated by an irreducible cyclic group, i.e. a group having one generator that has no non-trivial invariant subspace. We show how some of the basic properties of these codes, the cardinality and the minimum distance, can be derived using the isomorphism of the vector space and the extension field. Furthermore, we investigate the Pl\"ucker embedding of these codes and show how the orbit structure is preserved in the embedding.Comment: submitted to Designs, Codes and Cryptograph

Frequency permutation arrays

Motivated by recent interest in permutation arrays, we introduce and investigate the more general concept of frequency permutation arrays (FPAs). An FPA of length n=m lambda and distance d is a set T of multipermutations on a multiset of m symbols, each repeated with frequency lambda, such that the Hamming distance between any distinct x,y in T is at least d. Such arrays have potential applications in powerline communication. In this paper, we establish basic properties of FPAs, and provide direct constructions for FPAs using a range of combinatorial objects, including polynomials over finite fields, combinatorial designs, and codes. We also provide recursive constructions, and give bounds for the maximum size of such arrays.Comment: To appear in Journal of Combinatorial Design