Matematik Bölümünde 09.11.2015 tarihli seminer
Fen Fakültesi toplantı salonunda 9 Kasım 2015 Pazartesi günü saat 11:00’de Dr. Esengül Saltürk tarafından Codes over Local Frobenius Rings başlıklı matematik semineri verildi.
Codes over Local Frobenius Rings
Esengul Salturk, PhD
The theory of error correcting codes has been studied since 1948 due to Shannon's landmark paper
\A Mathematical Theory of Communication". While in classical coding theory everything was over nite
elds, in 1994, with the paper of Hammons and his collaborates, codes over nite rings have received
much attention. In many areas of coding theory, it is been standard to assume that the underlying
alphabet of a code is a nite Frobenius ring. This is because it is the largest class of rings for which the
full power of theory of codes apply.
It is well known that any nite commutative Frobenius ring is isomorphic to a direct product of local
Frobenius rings via the Chinese Remainder Theorem. This isomorphism carries many of the important
structural aspects of a code. Hence, there is a great signicance to studying codes over local Frobenius
Recently, local rings of order 16 have been classied by Martinez and Szabo in 2014. This talk covers
basics of error correcting codes and codes over local Frobenius rings of order 16 and their images to the
binary space via a Gray map which need not be linear. We dene Lee weight based on the Hamming
weight of the Gray image. This Gray map is a canonical extension of the usual Gray maps as dened
over local rings of order 4 and it is presented in a very general form which can be applied to any local
Frobenius ring of order 16 with a maximal ideal of size 8: We construct self-dual codes over these rings
and give their binary images.