Coding Rate Analysis of Forbidden Overlap Codes in High-Speed Buses

Abstract

One of the main problems in deep submicron designs of high-speed buses is propagation delay due to the crosstalk effect. To alleviate the crosstalk effect, there are several types of crosstalk avoidance codes proposed in the literature. In this article, we analyze the coding rates of forbidden overlap codes (FOCs) that avoid “010 → 101” transition and “101 → 010” transition on any three adjacent wires in a bus. We first compute the maximum achievable coding rate of FOCs and the maximum coding rate of memoryless FOCs. Our numerical results show that there is a significant gap between the maximum coding rate of memoryless FOCs and the maximum achievable rate. We then analyze the coding rates of FOCs generated from the bit-stuffing algorithm. Our worst-case analysis yields a tight lower bound of the coding rate of the bit-stuffing algorithm. Under the assumption of Bernoulli inputs, we use a Markov chain model to compute the coding rate of a bus with n wires under the bit-stuffing algorithm. The main difficulty of solving such a Markov chain model is that the number of states grows exponentially with respect to the number of wires n. To tackle the problem of the curse of dimensionality, we derive an approximate analysis that leads to a recursive closed-form formula for the coding rate over the nth wire. Our approximations match extremely well with the numerical results from solving the original Markov chain for n ⩽ 10 and the simulation results for n ⩽ 3000. Our analysis of coding rates of FOCs could be helpful in understanding the trade-off between propagation delay and coding rate among various crosstalk avoidance codes in the literature. In comparison with the forbidden transition codes (FTCs) that have shorter propagation delay than that of FOCs, our numerical results show that the coding rates of FOCs are much higher than those of FTCs.