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

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. I. INTRODUCTIONThe study of subblock-constrained codes has recently gained attention as they are suitable candidates for varied applications such as simultaneous energy and information transfer [1], powerline communications [2], and design of low-cost authentication methods [3]. A special class of subblock-constrained codes are codes where each codeword is partitioned into equal sized subblocks, and every subblock has the same fixed composition. Such codes were called constant subblock-composition codes (CSCCs) in [1], and were labeled as multiply constant-weight codesSubblock energy-constrained codes (SECCs) were proposed in [1] for providing real-time energy and information transfer from a powered transmitter to an energy harvesting receiver. For binary alphabet, SECCs are characterized A. Tandon and M. Motani are with the DRAFT 2 by the property that the weight of every subblock exceeds a given threshold. The CSCC and SECC capacities, and computable bounds, were presented in [1] for discrete memoryless channels. In this paper, we study bounds on the size and asymptotic rate for binary CSCCs and SECCs with given error correction capability, i.e., minimum distance of the code. A. NotationThe input alphabet is denoted X which comprises of q symbols. An n-length, q-ary code C over X is a subset of X n . The elements of C are called codewords and C is said to have distance d if the Hamming distance between any two distinct codewords is at least d. A q-ary code of length n and distance d is called an (n, d) q -code, and the largest size of an (n, d) q -code is denoted by A q (n, d). For binary alphabet (q = 2), an (n, d) 2 -code is just called an (n, d)-code, and the largest size for this code is simply denot...

Generalized Sphere-Packing Bound for Subblock-Constrained Codes

“…Nevertheless, we provide estimates to the optimization problem by judiciously choosing appropriate constrained spaces for estimating asymptotic ball sizes and subsquently, applying the generalized sphere-packing bound. Doing so, we extend the results in [6] to present improved upper bounds on the asymptotic rates for CSCCs and SECCs for a range of relative distance values.…”

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Wireless Power Transfer for Future Networks: Signal Processing, Machine Learning, Computing, and Sensing

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