In this paper we consider codes in Fs×rqwith packing radiusRregarding the NRT-metric (i.e. when the underlying poset is a disjoint union ofschains with the same lengthr) and we establish necessary condition on the parameterss,randRfor the existence of perfect codes. More explicitly, forr,s≥ 2 andR≥ 1 we prove that if there is a non-trivial perfect code then (r+ 1)(R+ 1) ≤rs. We also establish a correspondence between perfect codes withr>Rand those withr=R. Using this correspondence we prove the non-existence of non-trivial perfect codes in the casess≥R+ 2 ands= 3 over non-binary alphabet.

Let [ n { = {1, …, n } and let Ω n be the set of all mappings from [ n { to itself. Let f be a random uniform element of Ω n and let T( f ) and B( f ) denote, respectively, the least common multiple and the product of the length of the cycles of f . Harris proved in 1973 that T converges in distribution to a standard normal distribution and, in 2011, Schmutz obtained an asymptotic estimate on the logarithm of the expectation of T and B over all mappings on n nodes. We obtain analogous results for random uniform mappings on n = kr nodes with preimage sizes restricted to a set of the form {0,k}, where k = k ( r ) ≥ 2. This is motivated by the use of these classes of mappings as heuristic models for the statistics of polynomials of the form x k + a over the integers modulo p , with p ≡ 1 (mod k). We exhibit and discuss our numerical results on this heuristic.

In this paper, we establish some criteria to decide when a discrete memoryless channel admits a metric in such a way that the maximum likelihood decoding coincides with the nearest neighbor decoding. In particular, we prove a conjecture presented by M. Firer and J. L. Walker, establishing that every binary asymmetric channel admits a matched metric.

Let f be a uniformly random element of the set of all mappings from [n] = {1, ..., n} to itself. Let T(f) and B(f) denote, respectively, the least common multiple and the product of the lengths of the cycles of f. Harris proved in 1973 that log T converges in distribution to a standard normal distribution and, in 2011, Schmutz obtained an asymptotic estimate on the logarithm of the expectation of T and B over all mappings on n nodes. We obtain analogous results for uniform random mappings on n = kr nodes with preimage sizes restricted to a set of the form {0,k}, where k = k(r) >= 2. This is motivated by the use of these classes of mappings as heuristic models for the statistics of polynomials of the form x^k + a over the integers modulo p, where k divides p - 1. We exhibit and discuss our numerical results on this heuristic.

The Golomb-Welch conjecture (1968) on the non-existence of perfect Lee codes in Znwith radius e ≥ 2 and dimensions n ≥ 3, widely believed to be true, has been up to now only proved for large radius in any dimension, for small dimensions, and for some small radii and specific n. The main result of this paper is that for radius e = 2, there are no perfect Lee linear codes in Znfor infinitely many values of n.

We study the problem of characterizing when two memoryless binary asymmetric channels, described by their transition probabilities (p, q) and (p', q'), are equivalent from the point of view of maximum likelihood decoding when restricted to n-block binary codes. This equivalence of channels induces a partition (depending on n) on the space of parameters (p, q) into regions associated with the equivalence classes. Explicit expressions for describing these regions, their number and areas are derived. Some perspectives of applications of our results to decoding problems are also presented.

We describe the functional graph of the multiplication-by-$n$ map in a cycle group and use this to obtain the structure of the functional graph associated with a Rédei function over a nonbinary finite field $\mathbb{F}_q$. In particular, we obtain two descriptions of the tree attached to the cyclic nodes in these graphs and provide period and preperiod estimates for Rédei functions. We also extend characterizations of Rédei permutations by describing their decomposition into disjoint cycles. Finally, we obtain some results on the length of the cycles related to Rédei permutations and we give an algorithm to construct Rédei permutations with prescribed length cycles in a geometric progression.