On low-power error-correcting cooling codes with large distances

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 $e$ transmission 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.