Ivelisse Rubio

Cyclic Decomposition of Permutations of Finite Fields Obtained Using Monomials

In this paper we study permutations of finite fields F q that decompose as products of cycles of the same length, and are obtained using monomials \(x^i \in F_q[x]\). We give the necessary and sufficient conditions on the exponent i to obtain such permutations. We also present formulas for counting the number of this type of permutations. An application to the construction of encoders for turbo codes is also discussed.

Algebraic construction of interleavers using permutation monomials

We present an algebraic construction for interleavers of length p/sup r/, where p is any prime. These interleavers are very simple to implement and have performance better than random interleavers and other known algebraic constructions. We construct a permutation of /spl Zopf//sub p//sup r/ using permutations of the elements of the finite field F/sub p//sup r/ given by monomials over the field.

New families of balanced symmetric functions and a generalization of Cusick, Li and Stǎnicǎ’s conjecture

In this paper we provide new families of balanced symmetric functions over any finite field. We also generalize a conjecture of Cusick, Li, and Stǎnicǎ about the non-balancedness of elementary symmetric Boolean functions to any finite field and prove part of our conjecture.

Linear complexity for multidimensional arrays - a numerical invariant

Linear complexity is a measure of how complex a one dimensional sequence can be. In this paper we extend the concept of linear complexity to multiple dimensions and present a definition that is invariant under well-orderings of the arrays. As a result we find that our new definition for the process introduced in the patent titled “Digital Watermarking” produces arrays with good asymptotic properties.

Construction of systems of polynomial equations with exact p-Divisibility via the covering method

We present an elementary method to compute the exact p-divisibility of exponential sums of systems of polynomial equations over the prime field. Our results extend results by Carlitz and provide concrete and simple conditions to construct families of polynomial equations that are solvable over the prime field.

An Elementary Approach to Ax-Katz, McEliece's Divisibility and Applications to Quasi-Perfect Binary 2-Error Correcting Codes

In this paper we present an algorithmic approach to the problem of the divisibility of the number of solutions to a system of polynomial equations. Using this method we prove that all binary cyclic codes with two zeros over F2f and minimum distance 5 are quasi-perfect for f les 10. We also present elementary proofs of divisibility results that, in some cases, improve previous results

Exact 2-divisibility of exponential sums associated to boolean functions

In this paper we extend the covering method for computing the exact 2-divisibility of exponential sums of Boolean functions, improve results on the divisibility of the Hamming weight of deformations of Boolean functions, and provide criteria to obtain non-balanced functions. In particular, we present criteria to determine cosets of Reed-Muller codes that do not contain any balanced function, and to construct deformations of symmetric functions that are not balanced. The use of the covering method together with classifications of cosets of Reed-Muller codes obtained by the action of linear groups can improve the search of balanced functions in Reed-Muller codes dramatically.