Claudio Qureshi

Rédei Actions on Finite Fields and Multiplication Map in Cyclic Group

We describe the functional graph of the multiplication-by-

Codes and lattices in the l p metric

n

On perfect q-ary codes in the maximum metric

map in a cycle group and use this to obtain the structure of the functional graph associated with a Redei function over a nonbinary finite field

The graph structure of Chebyshev polynomials over finite fields and applications

\mathbb{F}_q

Cycle structure of iterating redei functions

. In particular, we obtain two descriptions of the tree attached to the cyclic nodes in these graphs and provide period and preperiod estimates for Redei functions. We also extend characterizations of Redei permutations by describing their decomposition into disjoint cycles. Finally, we obtain some results on the length of the cycles related to Redei permutations and we give an algorithm to construct Redei permutations with prescribed length cycles in a geometric progression.

The set of dimensions for which there are no linear perfect 2-error-correcting Lee codes has positive density.

Codes and associated lattices are studied in the l_p metric, particularly in the l₁ (Lee) and the l_∞ (maximum) distances. Discussions and results on decoding processes, classification and analysis of perfect or dense codes in these metrics are presented.

Matched Metrics to the Binary Asymmetric Channels.

Perfect codes in a given metric are the ones which attain the sphere packing bound. In the Hamming metric perfect linear codes over finite fields are known only to exist for a restricted number of parameters. In the Lee metric, perfect linear codes in Zn are classified for n = 2 and the long standing Golomb-Welch conjecture that there are no perfect codes for n ≥ 3, except for packing radius 1, still remains open. We approach here perfect codes over finite rings in the maximum metric (also known as L∞ or Chebyshev metric). This metric has been recently considered in coding schemes for flash memory. A perfect linear code in Znq in the maximum metric corresponds to an integer lattice cube tiling. We present a complete classification of two-dimensional perfect codes, constructions of perfect codes from codes of smaller dimensions and generator matrices for n-dimensional perfect linear q-ary codes in the maximum metric. From these results a description or all perfect codes in the Euclidean metric for n = 2, 3 is derived.

On the non-existence of linear perfect Lee codes: The Zhang-Ge condition and a new polynomial criterion.

We completely describe the functional graph associated to iterations of Chebyshev polynomials over finite fields. Then, we use our structural results to obtain estimates for the average rho length, average number of connected components and the expected value for the period and preperiod of iterating Chebyshev polynomials.