Perfect, Hamming and Simplex Linear Error-Block Codes with Minimum \(\pi \)-distance 3
Linear error-block codes were introduced in 2006 as a generalization of linear block codes. In this paper we construct two new families of perfect binary linear error-block codes of \( \pi \)-distance 3, namely, \([n_1]\ldots [n_t]^s\) (where \( t\ge 1 \)), and \([n_1][n_t]^s\) (where \( t= 1\) or \(t=2\)), we also introduce the notions of Hamming and Simplex linear error-block codes, and we give a method to construct Hamming LEB codes from its parity check matrix. We also prove that Hamming LEB codes are perfect, and the constructed perfect codes are Hamming.
KeywordsLinear error-block codes Simplex codes Hamming code Hamming bound and perfect codes
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