Matematik Bölümünde 09.11.2015 tarihli seminer

09.11.2015

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

Abstract

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 signi cance to studying codes over local Frobenius

rings.

Recently, local rings of order 16 have been classi ed 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 de ne Lee weight based on the Hamming

weight of the Gray image. This Gray map is a canonical extension of the usual Gray maps as de ned

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.