# Perfect, Hamming and Simplex Linear Error-Block Codes with Minimum $$\pi$$-distance 3

• Soukaina Belabssir
• Edoukou Berenger Ayebie
• El Mamoun Souidi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11445)

## Abstract

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, $$[n_1]\ldots [n_t][2]^s$$ (where $$t\ge 1$$), and $$[n_1][n_t][3]^s$$ (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.

## Keywords

Linear error-block codes Simplex codes Hamming code Hamming bound and perfect codes

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## Authors and Affiliations

• Soukaina Belabssir
• 1
• Edoukou Berenger Ayebie
• 1
• El Mamoun Souidi
• 1
Email author
1. 1.Faculty of Sciences, Laboratory of Mathematics, Computer Science, Applications and Information SecurityMohammed V University in RabatRabatMorocco