Let R=Z4 be the integer ring mod 4 and C be a linear code over R. The code C is called a triple cyclic code of length (r, s, t) over R if the set of its coordinates can be partitioned into three parts so that any cyclic shift of the coordinates of the three parts leaves the code invariant. These codes can be viewed as R[x]-submodules of R[x]/×R[x]/×R[x]/. In this paper, we determine the generator polynomials and the minimum generating sets of this kind of codes.

Generalized quasi-cyclic (GQC) codes with arbitrary lengths over the ring $mathbb{F}_{q}+umathbb{F}_{q}$, where u2=0, q=pn, n a positive integer and p a prime number, are investigated. By the Chinese Remainder Theorem, structural properties and the decomposition of GQC codes are given. For 1-generator GQC codes, minimum generating sets and lower bounds on the minimum distance are given.

In practical logic design circuits are built by composing certain types of gates. Each gate itself is a simple circuits with one, two or three inputs and one output, which implements an elementary logic function. These functions are called the generators. For the general purpose the set of generators is considered to be functionally complete, i. e. , it is able to express any logic function under chosen rules compositions. A basis is a functionally complete set of logic functions that contains no complete proper subset. Providing compactness and expressibility of the generators the notion of a basis, however, ignores the optimality of implementations. Efficiently irreducible generating set, termed ε-basis, is an irreducible set of generators which guarantees an optimal implementation of every function, with respect to the number of literals in its formal expression. The notion of ε-basis is significant in the composition of functions, since the classical definition of basis does not consider the efficiency of implementation. In case of Boolean functions, for two-input (dyadic) generators it has been shown that an ε-basis consists of all monadic functions, constants, and only two dyadic functions from certain classes. In this paper, expanding the domain of basic operations from dyadic to triadic, we study the efficiency of sets of 3-input gates as generators. This expansion decreases the complexity of functions (hence, the complexity of functional circuits to be designed). Gaining an evident merit in the complexity, we have to pay a price by a considerable increase of the number of such generators for the multiple valued circuits. However, in the case of Boolean operations this number is still very small, and it will certainly be useful to consider this approach in the practical circuit design. This paper provides a criterion for a generating set of triadic operations of k-valued logic to be efficiently irreducible. In the case of Boolean functions it is shown that there exist exactly five types of classes of triadic operations which constitute an ε-basis. A typical example of generator set which forms a triadic ε-basis, is also shown.