Bounds for codes for a non-symmetric ternary channel

In [1], a non-symmetric ternary communication channel inspired by 3-valued semiconductor memories was introduced, and error-correcting coding for this channel was studied. The authors of [1] showed the relevance of the minimum d1-distance (defined below) of a ternary code for judging its error-correcting capabilities on this channel, gave a code construction, and derived a Hamming-like upper bound on the size of a code of given length and minimum d1-distance. The work was extended in [2], where the authors obtained the channel capacity, and constructed optimal codes with a short length by techniques for finding cliques in graphs. In the present paper, we give upper and lower bounds on the size of codes for the d1distance. We first introduce some notation. We consider codes over the ternary alphabet Q = {−1, 0, 1}. For x,y ∈ Q, we define d1(x,y) as