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New characterisations of the Nordstrom–Robinson codes
Author(s) -
Gillespie Neil I.,
Praeger Cheryl E.
Publication year - 2017
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12016
Subject(s) - converse , transitive relation , mathematics , code (set theory) , combinatorics , discrete mathematics , binary number , arithmetic , computer science , set (abstract data type) , programming language , geometry
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.