Characterization of extended Hamming and Golay codes as perfect codes in poset block spaces

B. K. Dass; Namita Sharma; Rashmi Verma

doi:10.3934/amc.2018037

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2018, Volume 12, Issue 4: 629-639. Doi: 10.3934/amc.2018037

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$ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$

-additive cyclic codes

Characterization of extended Hamming and Golay codes as perfect codes in poset block spaces

1.
Department of Mathematics, University of Delhi, Delhi-110 007, India
2.
Mata Sundri College for Women (University of Delhi), Mata Sundri Lane, Delhi-110 002, India

* Corresponding author: Namita Sharma
* Corresponding author: Namita Sharma

Received: March 31, 2015

Revised: February 28, 2017

Published: October 2018

The first author is supported by a R & D grant of University of Delhi 2014-15. The second author is supported by Junior Research Fellowship Grant AA/139/F-177 of University Grants Commission

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Abstract

Alves, Panek and Firer (Error-block codes and poset metrics, Adv. Math. Commun., 2 (2008), 95-111) classified all poset block structures which turn the [8,4,4] extended binary Hamming code into a 1-perfect poset block code. However, the proof needs corrections that are supplied in this paper. We provide a counterexample to show that the extended binary Golay code is not 1-perfect for the proposed poset block structures. All poset block structures turning the extended binary and ternary Golay codes into 1-perfect codes are classified.

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Mathematics Subject Classification: Primary: 94B05; Secondary: 94B60.

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