Optimal linear identifying codes

Identifying codes can be used to locate malfunctioning processors. We say that a code C of length n is a linear (1,/spl les/l)-identifying code if it is a subspace of F/sub 2//sup n/ and for all X,Y/spl sube/F/sub 2//sup n/ such that |X|, |Y|/spl les/l and X/spl ne/Y, we have /spl cup//sub x/spl isin/X/(B(x)/spl cap/C)/spl ne//spl cup/y/spl isin/Y(B(y)/spl cap/C). Strongly (1,/spl les/l)-identifying codes are a variant of identifying codes. We determine the cardinalities of optimal linear (1,/spl les/l)-identifying and strongly (1,/spl les/l)-identifying codes in Hamming spaces of any dimension for locating any at most l malfunctioning processors.     

On algebraic decoding of the Z4-linear Goethals-likecodes

The Z₄-linear Goethals-like code of length 2^m has 2^2m+1-3m-2 codewords and minimum Lee distance 8 for any odd integer m⩾3. We present an algebraic decoding algorithm for all Z₄-linear Goethals-like codes C_k introduced by Helleseth et al.(1995, 1996). We use Dickson polynomials and their properties to solve the syndrome equations     

Two families of optimal identifying codes in binary Hamming spaces

A motivation for identifying codes comes from quality control in multiprocessor systems, that is, we are able, with the aid of these codes, to find faulty processors in such a system. We give a construction of two infinite families of optimal codes, which identify up to two malfunctioning processors in Hamming spaces