01/01/2006
We find a bound for the Goldie dimension of hereditary modules in terms of the cardinality of the generating sets of their quasiinjective hulls. Several consequences are deduced. In particular, it is shown that every finitely generated hereditary module with countably generated quasiinjective hull is noetherian. It is also shown that every right hereditary ring with finitely generated injective hull is right artinian, thus answering a long standing open question posed by Dung, Gómez Pardo and Wisbauer.
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01/01/2008
We investigate negacyclic and cyclic codes of length ps over the finite field Fpa. Negacyclic codes of length ps are precisely the ideals of the chain ring Fpa [x] / 〈xp^s+1〉. This structure is then used to obtain the Hamming distance distribution of the class of such negacyclic codes, which also provides Hamming weight distributions and enumerations of several codes. An onetoone correspondence between negacyclic and cyclic codes is established to carry accordingly those results of negacyclic codes to cyclic codes.
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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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01/01/2016
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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01/01/2013
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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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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01/01/2016
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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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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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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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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12/01/2005
Codes over the ring of integers modulo 4 have been studied by many researchers. Negacyclic codes such that the length n of the code is odd have been characterized over the alphabet ℤ4, and furthermore, have been generalized to the case of the alphabet being a finite commutative chain ring. In this paper, we investigate negacyclic codes of length 2s over Galois rings. The structure of negacyclic codes of length 2s over the Galois rings GR(2a, m), as well as that of their duals, are completely obtained. The Hamming distances of negacyclic codes over GR(2a, m) in general, and over ℤ2a in particular are studied. Among other more general results, the Hamming distances of all negacyclic codes over ℤ2a of length 4, 8, and 16 are given. The weight distributions of such negacyclic codes are also discussed.
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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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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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01/01/2007
Various kinds of distances of all negacyclic codes of length 2s over Zopf2a are completely determined. Using our structure theorems of negacyclic codes of length 2s over Zopf2a, we first calculate the Hamming distances of all such negacyclic codes, which particularly lead to the Hamming weight distributions and Hamming weight enumerators of several codes. These Hamming distances are then used to obtain their homogeneous, Lee, and Euclidean distances. Our techniques are extendable to the more general class of constacyclic codes, namely, the lambda constacyclic codes of length 2s over Zopf2a, where lambda is any unit of Zopf2a with the form 4k1. We establish the Hamming, homogeneous, Lee, and Euclidean distances of all such constacyclic codes.
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