# Generalized Sphere-Packing Bound for Subblock-Constrained Codes

## TL;DR

This paper applies the generalized sphere-packing bound to subblock-constrained codes, deriving closed-form bounds and improving asymptotic rate estimates for specific code classes.

## Contribution

It introduces a reduced-variable linear programming approach for bounds on CSCCs and SECCs, with closed-form solutions and asymptotic rate improvements.

## Key findings

- Closed-form bounds for single- and double-error CSCCs
- Improved asymptotic rate bounds for CSCCs and SECCs
- Numerical examples demonstrating bound improvements

## Abstract

We apply the generalized sphere-packing bound to two classes of subblock-constrained codes. A la Fazeli et al. (2015), we made use of automorphism to significantly reduce the number of variables in the associated linear programming problem. In particular, we study binary constant subblock-composition codes (CSCCs), characterized by the property that the number of ones in each subblock is constant, and binary subblock energy-constrained 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 single- and double-error correcting CSCCs. 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.