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.
Additive cyclic codes over Galois rings were investigated in @cite_4 . 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.
@cite_4
2565828551 1981604679 2258512292 1639538057
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. Let R = GR ( p ? , l ) be a Galois ring of characteristic p ? and cardinality p ? l , where p and l are prime integers. First, we give a canonical form decomposition for additive cyclic codes over R . This decomposition is used to construct additive cyclic codes and count the number of such codes, respectively. Then we give the trace dual code for each additive cyclic code over R from its canonical form decomposition and linear codes of length l over some extension Galois rings of Z p ? . In this paper we give a new sufficient condition for a general stability of Kronecker coefficients, which we call it additive stability. It was motivated by a recent talk of J. Stembridge at the conference in honor of Richard P. Stanley's 70th birthday, and it is based on work of the author on discrete tomography along the years. The main contribution of this paper is the discovery of the connection between additivity of integer matrices and stability of Kronecker coefficients. Additivity, in our context, is a concept from discrete tomography. Its advantage is that it is very easy to produce lots of examples of additive matrices and therefore of new instances of stability properties. We also show that Stembridge's hypothesis and additivity are closely related, and prove that all stability properties of Kronecker coefficients discovered before fit into additive stability. In this work, we present a new upper bound on the minimum distance d of linear locally repairable codes (LRCs) with information locality and availability. The bound takes into account the code length n, dimension k, locality r, availability t, and field size q. We use tensor product codes to construct several families of LRCs with information locality, and then we extend the construction to design LRCs with information locality and availability. Some of these codes are shown to be optimal with respect to their minimum distance, achieving the new bound. Finally, we study the all-symbol locality and availability properties of several classes of one-step majority-logic decodable codes, including cyclic simplex codes, cyclic difference-set codes, and 4-cycle free regular low-density parity-check (LDPC) codes. We also investigate their optimality using the new bound.