Additive cyclic codes over Galois rings were investigated in previous works. In this paper, we investigate the same problem but over a more general ring family, finite commutative chain rings. When we focus on non-Galois finite commutative chain rings, we observe two different kinds of additivity. One of them is a natural generalization of the previous studies, whereas the other one has some unusual properties especially while constructing dual codes. We interpret the reasons of such properties and illustrate our results giving concrete examples.
The first one, so-called Galois-additivity, is a natural generalization of the study in @cite_4 , anyway our way of construction in this generalization is slightly different from the one in that paper. The authors in @cite_4 were using some linear codes over the base ring and their generator matrices, but we just make use of ideals and do not get involved generator matrices. Our main result with this approach is Theorem . Also we have some further results related to the code size relations (Corollary ) and self-duality (Corollary ) as well as we illustrate these results in a concrete example (Example ).
@cite_4
1637999745 2565828551 2781245398 230449344
We describe a new class of list decodable codes based on Galois extensions of function fields and present a list decoding algorithm. These codes are obtained as a result of folding the set of rational places of a function field using certain elements (automorphisms) from the Galois group of the extension. This work is an extension of Folded Reed Solomon codes to the setting of Algebraic Geometric codes. We describe two constructions based on this framework depending on if the order of the automorphism used to fold the code is large or small compared to the block length. When the automorphism is of large order, the codes have polynomially bounded list size in the worst case. This construction gives codes of rate @math over an alphabet of size independent of block length that can correct a fraction of @math errors subject to the existence of asymptotically good towers of function fields with large automorphisms. The second construction addresses the case when the order of the element used to fold is small compared to the block length. In this case a heuristic analysis shows that for a random received word, the expected list size and the running time of the decoding algorithm are bounded by a polynomial in the block length. When applied to the Garcia-Stichtenoth tower, this yields codes of rate @math over an alphabet of size @math , that can correct a fraction of @math errors. Bierbrauer (2012) developed the theory of q-linear cyclic codes over (Fq)m and he obtained a parametric description of such codes by cyclotomic cosets. Recently, (2015) obtained the structure of cyclic additive codes over the Galois ring GR(p,m), where m is a prime integer. Let R be a finite commutative ring and Rn=R[x]xn1. In this paper, we generalize the theory of Fq-linear codes over vector spaces to R-linear codes over free R-algebras (free as R-module). We call these codes, R-additive codes. We introduce a one-to-one correspondence between the classes of cyclic R-additive code and the classes of Rn-linear code. Using the structure of Rn-linear codes, we present the structure of cyclic R-additive codes, where R is a chain ring. Among other results, q-linear cyclic codes over (Fq)m are described by ring-theoretic facts, and the structure of cyclic additive codes over the Galois ring GR(p,m) is given for an arbitrary integer m, not necessarily a prime number. We study various models of classical second-order set theories in the dependent type theory of Coq. Without logical assumptions, Aczel’s sets-as-trees interpretation yields an intensional model of second-order ZF with functional replacement. Building on work of Werner and Barras, we discuss the need for quotient axioms in order to obtain extensional models with relational replacement and to construct large sets. Specifically, we show that the consistency strength of Coq extended by excluded middle and a description operator on well-founded trees allows for constructing models with exactly n Grothendieck universes for every natural number n. By a previous categoricity result based on Zermelo’s embedding theorem, it follows that those models are unique up to isomorphism. Moreover, we show that the smallest universe contains exactly the hereditarily finite sets and give a concise independence proof of the foundation axiom based on permutation models. Let ( O ) k be the ring of integers of a finite extension k of the field ( Q ) p of p-adic numbers. The endomorphisms of a formal group law defined over ( O ) k provide nontrivial examples of commuting formal series with coefficients in ( O ) k . This article deals with the inverse problem formulated by Jonathan Lubin within the context of non-Archimedean dynamical systems. We present a large family of series, with coefficients in ( Z ) p , which satisfy Lubin's conjecture. These series are constructed with the help of Lubin–Tate formal group laws over ( Q ) p . We introduce the notion of minimally ramified series which turn out to be modulo p reductions of some series of this family. The commutant monoids of these minimally ramified series are determined by using the Fontaine–Wintenberger theory of the field of norms which allows an interpretation of them as automorphisms of ( Z ) p -extensions of local fields of characteristic zero. A particularly effective example illustrating the paper is given by a family of series generalizing Cebysev polynomials