Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes

Cyclic, negacyclic and constacyclic codes are part of a larger class of codes called polycyclic codes; namely, those codes which can be viewed as ideals of a factor ring of a polynomial ring. The structure of the ambient ring of polycyclic codes over GR(p^a,m) and generating sets for its ideals are considered. Along with some structure details of the ambient ring, the existance of a certain type of generating set for an ideal is proven.Comment: arXiv admin note: text overlap with arXiv:0906.400

The isomorphism problem for graph magma algebras

(One-value) graph magma algebras are algebras having a basis B = V boolean OR {1} such that, for all u,v is an element of V, uv is an element of {u,0}. Such bases induce graphs and, conversely, certain types of graphs induce graph magma algebras. The equivalence relation on graphs that induce isomorphic magma algebras is fully characterized for the class of associative graphs having only finitely many non-null connected components. In the process, the ring-the-oretic structure of the magma algebras induced by those graphs is given as it is shown that they are precisely those graph magma algebras that are semiperfect as rings. A complete description of the semiperfect rings that arise in this fashion, in ring theoretic and linear algebra terms, is also given. In particular, the precise number of isomorphism classes of one-value magma algebras of dimension n is shown to be Sigma(j <= n) p(j) where, for any i is an element of Z(+), p(i) is the number of partitions of i. While it is unknown whether uncountable dimensional algebras always have amenable bases, it is shown here that graph magma algebras do

Units and linear independence

An algebra A will be called fluid if for any linearly independent set X consisting of units, the set X−1 of inverses of the elements of X is also linearly independent. We consider various examples of fluid algebras and study fluid algebras in some specific settings including direct sums, field extensions, quotient rings of the ring of polynomials over a field, and matrix algebras. Pertinent parameters for the study of fluidity are also introduced and studied