New characterisations of the Nordstrom–Robinson codes

Neil Gillespie; Cheryl E. Praeger

doi:10.1112/blms.12016

New characterisations of the Nordstrom–Robinson codes

Neil Gillespie, Cheryl E. Praeger

Research output: Contribution to journal › Article (Academic Journal) › peer-review

3 Citations (Scopus)

269 Downloads (Pure)

Abstract

In his doctoral thesis, Snover proved that any binary (m, 256, δ) code is equivalent to the Nordstrom-Robinson code or the punctured Nordstrom-Robinson code for (m, δ) = (16, 6) or (15, 5) respectively. We prove that these codes are also characterised as completely regular binary codes with (m, δ) = (16, 6) or (15, 5), and moreover, that they are completely transitive. Also, it is known that completely transitive codes are necessarily completely regular, but whether the converse holds has up to now been an open question. We answer this by proving that certain completely regular codes are not completely transitive, namely, the (punctured) Preparata codes other than the (punctured) Nordstrom-Robinson code.

Original language	English
Pages (from-to)	320-330
Number of pages	11
Journal	Bulletin of the London Mathematical Society
Volume	49
Issue number	2
Early online date	9 Feb 2017
DOIs	https://doi.org/10.1112/blms.12016
Publication status	Published - 3 Apr 2017

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abstract = "In his doctoral thesis, Snover proved that any binary (m, 256, δ) code is equivalent to the Nordstrom-Robinson code or the punctured Nordstrom-Robinson code for (m, δ) = (16, 6) or (15, 5) respectively. We prove that these codes are also characterised as completely regular binary codes with (m, δ) = (16, 6) or (15, 5), and moreover, that they are completely transitive. Also, it is known that completely transitive codes are necessarily completely regular, but whether the converse holds has up to now been an open question. We answer this by proving that certain completely regular codes are not completely transitive, namely, the (punctured) Preparata codes other than the (punctured) Nordstrom-Robinson code.",

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