Generalized Sphere-Packing Bound for Subblock-Constrained Codes

We apply the generalized sphere-packing bound to two classes of subblock-constrained codes, in which each codeword is divided into smaller subblocks, and each subblock satisfies a certain application-dependent constraint. À la Fazeli et al. (2015), we make use of automorphisms to significantly reduce the number of variables in the associated linear programming problem. For the class of constant subblock-composition codes (CSCCs), we show that the optimization problem is equivalent to finding the minimum of N variables, where N is independent of the number of subblocks. We then provide closed-form solutions for the generalized sphere-packing bounds for single- and double-error correcting CSCCs. For the more general class of subblock energy-costrained codes (SECCs), we provide closed-form solutions for the generalized sphere-packing bounds for single errors in certain special cases.