Codes from zero-divisors and units in group rings

Codes from zero-divisors and units in group rings

We describe and present a new construction method for codes using encodings from group rings. They consist primarily of two types: zero-divisor and unit-derived codes. Previous codes from group rings focused on ideals; for example cyclic codes are ideals in the group ring over a cyclic group. The fresh focus is on the encodings themselves, which only under very limited conditions result in idea...

Smarandache-Zero Divisors in Group Rings

The study of zero-divisors in group rings had become interesting problem since 1940 with the famous zero-divisor conjecture proposed by G.Higman [2]. Since then several researchers [1, 2, 3] have given partial solutions to this conjecture. Till date the problem remains unsolved. Now we introduce the notions of Smarandache zero divisors (S-zero divisors) and Smarandache week zero divisors (S-wea...

Convolutional codes from units in matrix and group rings

A general method for constructing convolutional codes from units in Laurent series over matrix rings is presented. Using group rings as matrix rings, this forms a basis for in-depth exploration of convolutional codes from group ring encodings, wherein the ring in the group ring is itself a group ring. The method is used to algebraically construct series of convolutional codes. Algebraic methods...

Zero Divisors and Commutativity of Rings

Prüfer Conditions in Rings with Zero- Divisors

In his article: “Untersuchungen über die Teilbarkeitseigenschaften in Körpern” J. Reine Angew. Math. 168, 1 36, 1932 [21], Heinz Prüfer introduced a new class of integral domains, namely those domains R in which all finitely generated ideals are invertible. He also proved that to verify this condition, it suffices to check that it holds for all two-generated ideals of R. This was the modest beg...