New Bounds and Constructions for Constant Weighted X-Codes
Xiangliang Kong, Xin Wang, Gennian Ge · IEEE Transactions on Information Theory · 2021
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 introduced an equivalent definition of X-codes and studied X-codes of small weights that have good detectability and X-tolerance. An (m,n,d,x) X-code is an m× n binary matrix with column vectors as its codewords. The parametersd,xcorrespond to the test quality of the code. In this paper, bounds and constructions for constant weighted X-codes are investigated. First, we obtain a general result 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 the probabilistic method. Then, using tools from additive combinatorics and finite fields, we present some explicit constructions for constant weighted X-codes with d=3,7 and x=2, which are optimal for the case when d=3, w=4 and nearly optimal for the case when d=3, w=3. We also consider a special class of X-codes introduced by Fujiwara and Colbourn and improve the best known lower bound on the maximum number of codewords for this kind of X-codes.