8 research outputs found
New characterisations of the Nordstrom–Robinson codes
In his doctoral thesis, Snover proved that any binary code
is equivalent to the Nordstrom-Robinson code or the punctured
Nordstrom-Robinson code for or respectively. We
prove that these codes are also characterised as \emph{completely regular}
binary codes with or , and moreover, that they are
\emph{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
On the Structure of the Linear Codes with a Given Automorphism
The purpose of this paper is to present the structure of the linear codes
over a finite field with q elements that have a permutation automorphism of
order m. These codes can be considered as generalized quasi-cyclic codes.
Quasi-cyclic codes and almost quasi-cyclic codes are discussed in detail,
presenting necessary and sufficient conditions for which linear codes with such
an automorphism are self-orthogonal, self-dual, or linear complementary dual
Build a Sporadic Group in your Basement
All simple finite groups are classified as members of specific families. With one exception, these families are infinite collections of groups sharing similar structures. The exceptional family of sporadic groups contains exactly twenty-six members. The five Mathieu groups are the most accessible of these sporadic cases. In this article, we explore connections between Mathieu groups and error-correcting communication codes. These connections permit simple, visual representations of the three largest Mathieu groups: M24, M23, and M22. Along the way, we provide a brief, nontechnical introduction to the field of coding theory