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A construction of 2-quasi-perfect Lee codes is given over the space Z n p for p prime, p ≡ ±5 (mod 12) and n = 2[ p 4 ]. It is known that there are infinitely many such primes. Golomb and Welch conjectured that perfect codes for the Lee-metric do not exist for dimension n ≥ 3 and radius r ≥ 2. This conjecture was proved to be true for large radii as well as for low dimensions. The codes found are very close to be perfect, which exhibits the hardness of the conjecture. A series of computations show that related graphs are Ramanujan, which could provide further connections between Coding and Graph Theories.

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Abstract-Many current parallel computers are built around a torus interconnection network. Machines from Cray, HP, and IBM, among others, make use of this topology. In terms of topological advantages, square (2D) or cubic (3D) tori would be the topologies of choice. However, for different practical reasons, 2D and 3D tori with different number of nodes per dimension have been used. These mixed-radix topologies are not edge symmetric, which translates into poor performance due to an unbalanced use of network resources. In this work, we analyze twisted 2D and 3D mixed-radix tori that remove the network bottlenecks present in nontwisted ones. Such topologies recover edge symmetry, and consequently, balance the utilization of their links. The distance-related properties of twisted tori together with a full characterization of their bisection bandwidth are described in this paper. A simulation-based performance evaluation has been carried out to assess the network performance under synthetic and trace-driven workloads. The obtained results show noticeable and consistent performance gains (up to an increase of 74 percent in accepted load). In addition, we propose scalable and practicable packet routing mechanisms and wiring layouts for these interconnection systems. The complexity of the architectural proposals is similar to the one exhibited by routing and folding mechanisms in standard tori.

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