# Bounds on the Size and Asymptotic Rate of Subblock-Constrained Codes

## TL;DR

This paper derives bounds on the size and asymptotic rate of subblock-constrained codes, specifically CSCCs and SECCs, revealing their rate relationships and correcting previous results with numerical comparisons.

## Contribution

It introduces novel bounds for binary CSCCs and SECCs, compares their asymptotic rates, and corrects earlier findings in the literature.

## Key findings

- Asymptotic rate for CSCCs is lower than for CWCs at small relative distances.
- Asymptotic rate for SECCs is lower than for HWCs at high weight and low relative distance.
- Corrects a previous result by Chee et al. (2014) on CSCC rate.

## Abstract

The study of subblock-constrained codes has recently gained attention due to their application in diverse fields. We present bounds on the size and asymptotic rate for two classes of subblock-constrained codes. The first class is binary constant subblock-composition codes (CSCCs), where each codeword is partitioned into equal sized subblocks, and every subblock has the same fixed weight. The second class is binary subblock energy-constrained codes (SECCs), where the weight of every subblock exceeds a given threshold. We present novel upper and lower bounds on the code sizes and asymptotic rates for binary CSCCs and SECCs. For a fixed subblock length and small relative distance, we show that the asymptotic rate for CSCCs (resp. SECCs) is strictly lower than the corresponding rate for constant weight codes (CWCs) (resp. heavy weight codes (HWCs)). Further, for codes with high weight and low relative distance, we show that the asymptotic rates for CSCCs is strictly lower than that of SECCs, which contrasts that the asymptotic rate for CWCs is equal to that of HWCs. We also provide a correction to an earlier result by Chee et al. (2014) on the asymptotic CSCC rate. Additionally, we present several numerical examples comparing the rates for CSCCs and SECCs with those for constant weight codes and heavy weight codes.