Line Codes Generated by Finite Coxeter Groups

Abstract

Using an algebraic approach based on the theory of Coxeter groups, we design, and describe the performance of, a class of line codes derived from permutation modulation, useful for parallel transmission of $b$ bits over $b+1$ wires, and admitting especially simple encoding and decoding algorithms. With these codes, resistance to common-mode noise is obtained by using codewords whose components sum to zero, simultaneous switching output noise is reduced by using constant-energy signals, and the effects of intersymbol interference are reduced by having decisions based on only two values at the input of the final slicers. Codebook design is based on the theory of Group Codes for the Gaussian Channel, as specialized to Coxeter matrix groups generated by reflections in orthogonal hyperplanes. A number of designs are exhibited, some of them being novel or improving on previously obtained codes.