Bounds on r-identifying codes in q-ary Lee space
DOI:
https://doi.org/10.11575/cdm.v16i1.62666Abstract
Identifying codes are used to locate malfunctioning processors in multiprocessor systems. In this paper, we study identifying codes in a $q$-ary hypercube which is used in parallel processing. Computing upper and lower bounds of $M_{r,q}(n),$ the smallest cardinality among all $r$-identifying codes in $\mathbb{Z}_q^n$ with respect to the Lee metric, is an important research problem in this area. Using our constructions, we produce tables for upper and lower bounds for $M_{r,q}(n)$. The upper and the lower bounds of $M_{r,4}(n)$ known only when $r=1$ but using our results, we compute the bounds for $M_{r,4}(n)$ for all $r\geq 1$. Also we improve upon the currently known upper bounds of $M_{1,4}(n)$ due to J. L. Kim and S. J. Kim. Upper bounds of $M_{r,q}(n)$ for $q>4$ are known previously for some cases of $n$. We improve some of these bounds and we also compute bounds for all $n$ by using our results.
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