Set Shaping Theory and the Foundations of Redundancy-Free Testable Codes

Abstract: To render a sequence testable, namely capable of identifying and detecting errors, it is necessary to apply a transformation that increases its length by introducing statistical dependence among symbols, as commonly exemplified by the addition of parity bits. However, since the decoder does not have prior knowledge of the original symbols, it must treat the artificially introduced symbols as if they were independent. Consequently, these additional symbols must be transmitted, even though their conditional probability, under ideal and error free conditions, would be zero. This sequence extension implies that not all symbol combinations of the new length are practically realizable: if an error modifies a sequence, making it inadmissible such an error becomes detectable. Recent developments in Set Shaping Theory have revealed a surprising result: it is always possible to transform a sequence into a longer version by carefully selecting which longer sequences are allowed, in such a way that the overall set of sequences becomes more structured and less complex than the original. This means that even though the sequence is extended and dependencies are introduced between symbols, the total amount of information contained in the new set does not increase proportionally on the contrary, it can be slightly reduced. In other words, one can construct a new set of longer sequences where each one corresponds uniquely to an original sequence, but the entire set is designed in such a way that it can be treated as if the symbols were independent, making encoding simpler. This allows sequence to become testable capable of detecting errors without adding visible redundancy or increasing the informational content.