Let R be a Dedekind ring, K its quotient field, L a separable finite extension over K , and O L the integral closure of R in L. In this paper we provide a "practical" criterion that tests when a given α ∈ O L generates a power basis for O L over R (i.e. when O L = R[α]), improving significantly a result in this direction by M. Charkani and O. Lahlou. Applications in the context of cyclotomic, quadratic, biquadratic number fields, and some Dedekind rings are provided.
A well-known lower bound (over finite fields and some special finite commutative rings) on the Hamming distance of a matrix-product code (MPC) is shown to remain valid over any commutative ring R. A sufficient condition is given, as well, for such a bound to be sharp. It is also shown that an MPC is free when its input codes are all free, in which case a generating matrix is given. If R is finite, a sufficient condition is provided for the dual of an MPC to be an MPC, a generating matrix for such a dual is given, and characterizations of LCD, self-dual, and self-orthogonal MPCs are presented. Finally, results of this paper are used along with previous results of the authors to construct novel MPCs arising from (σ, δ)-codes. Some properties of such constructions are also studied.
In this note, we show that the result [1, Proposition 5.2] is inaccurate. We further give and prove the correct modification of such a result. Some applications are also given.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.