Use of Set Shaping theory in the development of locally testable codes

TL;DR

This paper introduces a novel approach to locally testable codes by applying Set Shaping Theory to add redundancy before encoding, enabling efficient testing and error detection based on conditional probabilities.

Contribution

It proposes using Set Shaping Theory to introduce redundancy pre-encoding, offering an alternative to traditional codeword redundancy for local testability.

Findings

The method allows error detection through zero conditional probability symbols.

Codes designed with minimal average information content are tested efficiently.

Longer strings are balanced by lower entropy, improving compression and testability.

Abstract

In developing locally testable codes, information is added to the coded message by creating redundancy in the codewords. In this article, we propose an alternative method in which redundancy is introduced on the message that must be transmitted before its encoding and not in the codewords. This approach exploits the Set Shaping Theory whose goal is the study the bijection functions f(X)=Y that transform a set of strings into a set of equal size made up of strings of greater length. In this way, this type of function transforms the independent variable x into the dependent variable y whose emission probability is conditioned by the previously emitted variables. Thus, if the decoder decodes a symbol associated with a conditional probability equal to zero, we detect an error in the message. If the function f used is the one that minimizes the average information content, we develop a code that can be tested efficiently. In fact, it is observed, in terms of compression, that the greater length of the strings is compensated by the fact of having chosen the strings with less entropy.