X-Codes: Theory and Applications of Unknowable Inputs

This paper studies the properties of a new class of codes introduced recently (and currently being used) in the context of digital system test compaction [12]. These codes, named X-codes after the “X” symbol used to denote unknown logic values in digital systems, address the problem of detecting errors in the presence of unknowns. Specifically, an X-code produces a hash (or signature) over a set of input bits in a way that guarantees that errors in the inputs lead to changes in the hash despite the presence of unknown inputs. In contrast with erasure-based approaches, X-codes assume that operations (e.g., solution of linear equations) to calculate the values of unknown inputs are impractical or impossible. In effect, the input values are unknowable. These properties do not allow characterization in terms of Hamming distance alone, and, to the best of our knowledge, X-codes have not been studied prior to their recent introduction. The X-code may also be useful for other applications, such as communications, but our primary focus is currently on digital system designs. For the purposes of this paper, we restrict our discussion to binary linear X-codes, although X-codes could be generalized in the future. The paper is structured as follows. The next section describes the digital system testing problem and the practical issues that give rise to X-codes. In later sections, we develop a mathematical formulation for the codes, study the relationships between different classes of X-codes, discuss a few structural elements of X-code matrices, and compare them with the superimposed codes developed for combinatorial group testing (CGT). We next describe several constructions, then conclude with a few asymptotic bounds.