r-Identifying codes in binary Hamming space, q-ary Lee space and incomplete hypercube

We study about monotonicity of r-identifying codes in binary Hamming space, q-ary Lee space and incomplete hypercube. Also, we give the lower bounds for M-1,q((<= l)) (n) where M-1,q((<= l)) (n) is the smallest cardinality among all r-identifying codes in Z(q)(n) with respect to the Lee metric. We prove the existence of 1-identifying code in an incomplete hypercube. Also, we give the construction techniques for r-identifying codes in the incomplete hypercubes in Secs. 4.1 and 4.2. Using these techniques, we give the tables (see Tables 1-6) of upper bounds for M-IH,M-r(k) where M-IH,M-r(k) is the smallest cardinality among all r-identifying codes in an incomplete hypercube with k processors. Also, we give the exact values of M-IH,M-r(k) for small values of r and k (see Sec. 4.3).