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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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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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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06/01/2012
Let Q be the field of rational numbers. As a module over the ring Z of integers, Q is Zprojective, but QZ is not a projective module. Contrary to this situation, we show that over a prime right noetherian right hereditary right Vring R, a right module P is projective if and only if P is Rprojective. As a consequence of this we obtain the result stated in the title. Furthermore, we apply this to affirmatively answer a question that was left open in a recent work of Holston, LópezPermouth and Orhan Ertag (2012) by showing that over a right noetherian prime right SIring, quasiprojective right modules are projective or semisimple.
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01/01/2012
The algebraic structures in term of polynomial generators of all constacyclic codes of length 2ps over the finite field Fpm are established. Among other results, all selfdual negacyclic codes of length 2ps, where p ≡ 1 (mod 4) (any m), or p ≡ 3 (mod 4) and m is even, are provided. It is also shown the nonexistence of selfdual negacyclic codes of length 2ps, where p ≡ 3 (mod 4), m is odd, and selfdual cyclic codes of length 2ps, for any odd prime p.
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09/01/2010
For any prime p, all constacyclic codes of length ps over the ring ℛ = Fpm + uFpm are considered. The units of the ring ℛ are of the forms γ and ⍺ + uβ, where ⍺, β, and γ are nonzero elements of Fpm, which provides pm (pm 1) such constacyclic codes. First, the structure and Hamming distances of all constacyclic codes of length ps over the finite field Fpm are obtained; they are used as a tool to establish the structure and Hamming distances of all (⍺ + uβ)constacyclic codes of length ps over ℛ. We then classify all cyclic codes of length ps over ℛ and obtain the number of codewords in each of those cyclic codes. Finally, a onetoone correspondence between cyclic and γconstacyclic codes of length ps over ℛ is constructed via ring isomorphism, which carries over the results regarding cyclic codes corresponding to γconstacyclic codes of length ps over ℛ.
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04/01/2009
We study all constacyclic codes of length 2s over GR(Rfr,m), the Galois extension ring of dimension m of the ring Rfr=F2+uF2. The units of the ring GR(Rfr,m) are of the forms alpha, and alpha+ubeta, where alpha, beta are nonzero elements of F2m, which correspond to 2 m(2m1) such constacyclic codes. First, the structure and Hamming distances of (1+ugamma)constacyclic codes are established. We then classify all cyclic codes of length 2sover GR(Rfr,m), and obtain a formula for the number of those cyclic codes, as well as the number of codewords in each code. Finally, onetoone correspondences between cyclic and alphaconstacyclic codes, as well as (1+ugamma)constacyclic and (alpha+ubeta) constacyclic codes are provided via ring isomorphisms, that allow us to carry over the results about cyclic and (1+ugamma)constacyclic accordingly to all constacyclic codes of length 2s over GR(Rfr,m).
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