2022/02/21 by E. J. García-Claro, García-Claro, E. J., I. S. Gutiérrez +1 · 1 citation
Computer Science · Engineering · #Coding theory and cryptography #Cooperative Communication and Network Coding #FOS: Computer and information sciences #Information Theory (cs.IT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2202.10005
openalex publication_date 2022/02/21 · openalex created_date 2022/07/13 · openalex updated_date 2026/07/28
Generating functions for the size of a r-sphere, with respect to the Manhattan distance in an n-dimensional grid, are used to provide explicit formulas for the minimum and maximum size of an r-ball centered at a point of the grid. This allows us to offer versions of the Hamming and Gilbert-Varshamov bounds for codes in these grids. Relations between the Hamming, Manhattan, and Lee distances defined in an abelian group G are studied. A formula for the minimum Hamming distance of codes that are cyclic subgroups of G is presented. Furthermore, several lower bounds for the minimum Manhattan distance of these codes based on their minimum Hamming and Lee distances are established. Examples illustrating the main results are presented, including several SageMath implementations.