Identifying Codes in q-ary Hypercubes

Let q be any integer ≥ 2. In this paper, we consider the q-ary ndimensional cube whose vertex set is Znq and two vertices (x1, . . . , xn) and (y1, . . . , yn) are adjacent if their Lee distance is 1. As a natural extension of identifying codes in binary Hamming spaces, we further study identifying codes in the above q-ary hypercube. We let M t (n) denote the smallest cardinality of t-identifying codes of length n in Z n q . Little is known about ternary or quaternary identifying codes. It is known [2, 14] that M 1 (n) ≥ 2v d+1+ 2 n where v is the number of vertices of Zn2 and d is the degree of any vertex of Z n 2 . In a similar manner, we show that M 1 (n) ≥ 2v d+1+ 1 n , where d is the degree and v = v(q) is the number of vertices of Znq for q = 3 and q = 4, respectively. We also give some constructions to show that M 1 (2) = 4, M 1 (3) = 9, and M 4 1 (2) = 7, deriving some upper bounds on M 1 (n) and M 4s 1 (n).