Linear programming-based optimization of the distance spectrum of linear block codes

We describe an approach for the identification of good distance spectra for possibly existing binary linear block codes based on linear programming and the MacWilliams-Delsarte (1977, 1972) identities. Specifically, the linear program is defined by an expression characterizing the performance of a potential code in terms of its distance spectrum and constraints imposed by the MacWilliams-Delsarte identities. Using the union bound to characterize performance, our results suggest that the best distance spectrum is not a function of signal-to-noise ratio (SNR) above the cutoff rate SNR and also suggest the existence of several unknown, good codes. Characterizing the performance using the maximum spectral error component of the union bound suggests spectral thinning with decreasing SNR.