New bounds and constructions for constant weighted X-codes
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As a crucial technique for integrated circuits (IC) test response compaction, X-compact employs a special kind of codes called X-codes for reliable compressions of the test response in the presence of unknown logic values (Xs). From a combinatorial view point, Fujiwara and Colbourn FC2010 introduced an equivalent definition of X-codes and studied X-codes of small weights that have good detectability and X-tolerance. In this paper, bounds and constructions for constant weighted X-codes are investigated. First, we prove a general lower bound on the maximum number of codewords n for an (m,n,d,x) X-code of weight w, and we further improve this lower bound for the case with x=2 and w=3 through a probabilistic hypergraph independent set approach. Then, using tools from additive combinatorics and finite fields, we present some explicit constructions for constant weighted X-codes with d=3,5 and x=2, which are nearly optimal for the case when d=3 and w=3. We also consider a special class of X-codes introduced in FC2010 and improve the best known lower bound on the maximum number of codewords for this kind of X-codes.
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