On Low-Power Error-Correcting Cooling Codes With Large Distances

Abstract

A low-power error-correcting cooling (LPECC) code was introduced as a coding scheme for communication over a bus by Chee et al. to control the peak temperature, the average power consumption of on-chip buses, and error-correction for the transmitted information, simultaneously. Specifically, an $(n, t, w, e)$ -LPECC code is a coding scheme over n wires that avoids state transitions on the t hottest wires and allows at most w state transitions in each transmission, and can correct up to etransmission errors. In this paper, we study the maximum possible size of an $(n, t, w, e)$ -LPECC code, denoted by $C(n,t,w,e)$ . When $w=e+2$ is large, we establish a general upper bound $C(n,t,w,w-2)\leq \lfloor \binom {n+1}{2}/\binom {w+t}{2}\rfloor $ ; when $w=e+2=3$ , we prove $C(n,t,3,1) \leq \lfloor \frac {n(n+1)}{6(t+1)}\rfloor $ . Both bounds are tight for large n satisfying some divisibility conditions. Previously, tight bounds were known only for $w=e+2=3,4$ and $t\leq 2$ . In general, when $w=e+d$ is large for a constant d, we determine the asymptotic value of $C(n,t,w,w-d)\sim \binom {n}{d}/\binom {w+t}{d}$ as n goes to infinity, which can be extended to q-ary codes.