Rate-Constrained Shaping Codes for Finite-State Channels With Cost

TL;DR

This paper investigates rate-constrained shaping codes for finite-state channels with costs, establishing optimality conditions and a separation theorem that combines compression and shaping for minimal average cost.

Contribution

It introduces a general framework for shaping codes on finite-state channels, linking rate and cost, and proves a separation theorem for optimal shaping strategies.

Findings

Derived relationship between code rate and average symbol cost.

Identified the rate that minimizes total average cost.

Proved a separation theorem for optimal shaping via compression and shaping.

Abstract

Shaping codes are used to generate code sequences in which the symbols obey a prescribed probability distribution. They arise naturally in the context of source coding for noiseless channels with unequal symbol costs. Recently, shaping codes have been proposed to extend the lifetime of flash memory and reduce DNA synthesis time. In this paper, we study a general class of shaping codes for noiseless finite-state channels with cost and i.i.d. sources. We establish a relationship between the code rate and minimum average symbol cost. We then determine the rate that minimizes the average cost per source symbol (total cost). An equivalence is established between codes minimizing average symbol cost and codes minimizing total cost, and a separation theorem is proved, showing that optimal shaping can be achieved by a concatenation of optimal compression and optimal shaping for a uniform i.i.d. source.