Lee codes have been intensively studied for more than 40 years. Interest in these codes has been triggered by the Golomb-Welch conjecture on the existence of the perfect error-correcting Lee codes. In this paper we deal with the existence and enumeration of diameter perfect Lee codes. As main results we determine all $q$ for which there exists a linear diameter-4 perfect Lee code of word length $n$ over $Z_{q},$ and prove that for each $n\geq 3$ there are uncountable many diameter-4 perfect Lee codes of word length $n$ over $Z.$ This is in a strict contrast with perfect error-correcting Lee codes of word length $n$ over $Z\,$\ as there is a unique such code for $n=3,$ and its is conjectured that this is always the case when $2n+1$ is a prime. We produce diameter perfect Lee codes by an algebraic construction that is based on a group homomorphism. This will allow us to design an efficient algorithm for their decoding. We hope that this construction will turn out to be useful far beyond the scope of this paper

An n-dimensional cross consists of 2n + 1 unit cubes: the "central" cube and reflections in all its faces. A tiling by crosses is called a Z-tiling if each cross is centered at a point with integer coordinates. Periodic tilings of R n by crosses have been constructed by several authors for all n ∈ N . No non-periodic tiling of R n by crosses has been found so far. We prove that if 2n + 1 is not a prime, then the total number of non-periodic Z-tilings of R n by crosses is 2 ℵ 0 while the total number of periodic Z-tilings is only ℵ 0 . In a sharp contrast to this result we show that any two tilings of R n , n = 2, 3, by crosses are congruent. We conjecture that this is the case not only for n = 2, 3, but for all n where 2n + 1 is a prime.

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Using group theory approach, we determine all numbers q for which there exists a linear 1-error correcting perfect Lee code of block length n over Z q , and then we enumerate those codes. At the same time this approach allows us to design a linear time decoding algorithm. Mathematics Subject Classifications (2000) 94B35 · 05E20A code C is a subset of a metric space (M, ρ); C is a perfect e-error correcting code if for every word V ∈ M there is exactly one codeword W ∈ C so that ρ(V, W ) ≤ e.The most common metric in coding theory is the Hamming metric. In this paper we deal with another frequently used metric, the so called Lee metric. In this case C is called a Lee code. As usual, let Z be the set of all integers, Z q denote the ring of integers modulo q, and let T n stand for the n-fold Cartesian product of a set T . First we consider Lee codes when M = Z n q . The Lee metric ρ L (U, V ), where U, V ∈ Z n q , U = (u 1 , u 2 , . . . , u n ),. Such a Lee code is called a Lee code of block size n over Z q . The perfect e-error correcting Lee code of block size n over Z q , where q ≥ 2e + 1, will be denoted by P L(n, e, q).

In this paper, we use a pseudo-Boolean formulation of the p-median problem and using data aggregation, provide a compact representation of p-median problem instances. We provide computational results to demonstrate this compactification in benchmark instances. We then use our representation to explain why some p-median problem instances are more difficult to solve to optimality than other instances of the same size. We also derive a preprocessing rule based on our formulation, and describe equivalent p-median problem instances, which are identical sized instances which are guaranteed to have identical optimal solutions.