06/01/2016
The units of the chain ring ℛa = Fpm [u]/〈ua〉 = Fpm + uFpm + ⋯ + ua−1Fpm are partitioned into a distinct types. It is shown that for any unit Λ of Type k, a unit λ of Type k∗ can be constructed, such that the class of λconstacyclic of length ps of Type k∗ codes is onetoone correspondent to the class of Λconstacyclic codes of the same length of Type k via a ring isomorphism. The units of ℛa of the form Λ = Λ0 + u Λ1 + ⋯ + ua−1 Λa−1, where Λ0, Λ1, … , Λa−1 ∈ Fpm, Λ0 ≠ 0, Λ1 ≠ 0, are considered in detail. The structure, duals, Hamming and homogeneous distances of Λconstacyclic codes of length ps over ℛa are established. It is shown that selfdual Λconstacyclic codes of length ps over ℛa exist if and only if a is even, and in such case, it is unique. Among other results, we discuss some conditions when a code is both α and βconstacyclic over ℛa for different units α, β.
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01/01/2016
This paper overviews the study of skew Θλconstacyclic codes over finite fields and finite commutative chain rings. The structure of skew Θλconstacyclic codes and their duals are provided. Among other results, we also consider the Euclidean and Hermitian dual codes of skew Θcyclic andskew Θnegacyclic codes over finite chain rings in general and over Fpm + uFpm in particular. Moreover, general decoding procedure for decoding skew BCH codes with designed distance and an algorithm for decoding skew BCH codes are discussed.
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01/01/2016
The aim of this paper is to determine the algebraic structures of all λconstacyclic codes of length 2 p s over the finite commutative chain ring F p m + u F p m , where p is an odd prime and u 2 = 0 . For this purpose, the situation of λ is mainly divided into two cases separately. If the unit λ is not a square and λ = α + u β for nonzero elements α , β of F p m , it is shown that the ambient ring ( F p m + u F p m ) x / { x 2 p s  ( α + u β ) } is a chain ring with the unique maximal ideal { x 2  α 0 } , and thus ( α + u β ) constacyclic codes are { ( x 2  α 0 ) i } for 0 ≤ i ≤ 2 p s . If the unit λ is not a square and λ = γ for some nonzero element γ of F p m , such λconstacyclic codes are classified into 4 distinct types of ideals. The detailed structures of ideals in each type are provided. Among other results, the number of codewords and the dual of every λconstacyclic code are obtained.
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09/01/2015
For any given positive integer m, a necessary and sufficient condition for the existence of TypeI madic constacyclic codes is given. Furthermore, for any given integer s, a necessary and sufficient condition for s to be a multiplier of a TypeI polyadic constacyclic code is given. As an application, some optimal codes from TypeI polyadic constacyclic codes, including generalized ReedSolomon codes and alternant maximum distance separable codes, are constructed.
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05/01/2015
For any different odd primes ℓ and p, structure of constacyclic codes of length 2ℓmpn over the finite field Fq of characteristic p and their duals is established in terms of their generator polynomials. Among other results, the characterization and enumeration of all linear complementary dual and selfdual constacyclic codes of length 2ℓmpn are obtained.
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11/20/2014
An equivalence relation is introduced on the nonzero elements of the finite field Fpm to classify constacyclic codes of arbitrary length over Fpm. According to the equivalence classes, all constacyclic codes of length ℓps over Fpm and their duals are characterized, where ℓ is a prime different from p and s is a positive integer. Selfdual cyclic codes of length ℓps over Fpm exist precisely when p is equal to two; in this case, all selfdual cyclic codes of length 2sℓ over F2m are presented.
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09/01/2014
This note gives a counterexample of Theorem 20 in the paper of Blackford (2013) [2]. The counterexample shows that [2, Theorem 20] is incorrect. Furthermore, we provide corrections to the above result.
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02/01/2014
This work is concerned with the abstract Cauchy problems that depend on parameters. The goal is to study continuity in the parameters of the classical solutions of the Cauchy problems. The situation considered in this work is when the operator of the Cauchy problem is not densely defined. By applying integrated semigroup theory and the results on continuity in the parameters ofC0semigroup and integrated semigroup, we obtain the results on the existence and continuity in parameters of the classical solutions of the Cauchy problems. The application of the obtained abstract results in a parabolic partial differential equation is discussed in the last section of the paper.
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05/06/2013
Let p≠3 be any prime. A classification of constacyclic codes of length 3ps over the finite field Fpm is provided. Based on this, the structures in terms of polynomial generators of all such constacyclic codes and their duals are established. Among other results, we show that selfdual cyclic codes of length 3ps exist only when p=2, and in such case, those selfdual codes are listed.
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01/01/2013
A ring R is called a right weakly Vring (briefly, a right WVring) if every simple right Rmodule is Xinjective, where X is any cyclic right Rmodule with XR ≇ RR. In this note, we study the structure of right WVrings R and show that, if R is not a right Vring, then R has exactly three distinct ideals, 0 ⊂ J ⊂ R, where J is a nilpotent minimal right ideal of R such that R/J is a simple right Vdomain. In this case, if we assume additionally that RJ is finitely generated, then R is left Artinian and right uniserial with composition length 2. We also show that a strictly right WVring with Jacobson radical J is a Frobenius local ring if and only if the injective hull of JR is uniserial. Some other results are obtained in the connection with the Noetherian property of right WVrings and related rings.
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