Generalized Sphere-Packing Bound for Subblock-Constrained Codes

We apply the generalized sphere-packing bound to two classes of subblock-constrained codes. <italic>À la</italic> Fazeli <italic>et al.</italic> (2015), we make use of automorphisms to significantly reduce the number of variables in the associated linear programming problem. In particular, we study binary <italic>constant subblock-composition</italic> codes (CSCCs), characterized by the property that the number of ones in each subblock is constant, and binary <italic>subblock energy-constrained</italic> codes (SECCs), characterized by the property that the number of ones in each subblock exceeds a certain threshold. For 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 t-error correcting CSCCs for <inline-formula> <tex-math notation="LaTeX">$\text {t}\in \{1,2,3\}$ </tex-math></inline-formula>. For SECCs, we provide closed-form solutions for the generalized sphere-packing bounds for single errors in certain special cases. We also obtain improved bounds on the optimal asymptotic rate for CSCCs and SECCs, and provide numerical examples to highlight the improvement.