06/01/2016
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04/06/2016
When communicating across a channel, it is inevitable that such pathways of communication be “noisy”, thus there is always some sort of interference across the channel. This results in messages not always being received as they were sent. In order to solve these problems, coding theory developed and is used both to detect and correct errors. It is used for data compression, error correction, cryptography and network coding. In error correction, a concentration on algebraic coding theory lies with linear codes, including cyclic and constacyclic codes. In this poster presentation, we will discuss the history of coding theory, going in depth with cyclic and constacyclic codes, as well as discussing applications and current problems being resolved using algebraic coding theory.
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03/15/2016
In this work, we shall investigate error correction in binary messages of finite arbitrary length in order to find the densest packing of codewords, i.e., the most efficient set of codewords. This is an important area of research because of the increasing reliance on digital communications, which are based on binary messages. We build on the work of Richard Hamming who contributed the Hamming distance, which is a method for measuring the difference between two segments, i.e., words, of a message. Our results suggest that for a message of length n, where n is at least 4, there are 2^(n3) singleerror correctable codewords out of 2^n possible words. These results will allow us to make further strides toward finding efficient multipleerror correctable codes in the future.
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03/15/2016
Coding theory has developed alongside the use of communication. When communicating across a channel, it is inevitable that such pathways of communication be “noisy”. The absence of noise is virtually impossible to attain, thus there is always some sort of interference across the communication channel. This results in messages not always being received as they were originally sent. In order to solve these problems, coding theory developed. It is used both to detect and correct errors in a variety of codes. Coding theory has grown into a discipline affecting all sorts of areas including but not limited to computer science, mathematics, and engineering. The codes can be used for data compression (or source coding), error correction (or channel coding), cryptography and even network coding. Primarily in error correction do we find codes which we are able to transmit quickly, contain many codewords, and also can correct or detect many errors. Within this discipline, a concentration on algebraic coding theory lies with linear codes. There are a variety of different codes discovered within this set of linear codes, including cyclic and constacyclic codes. In this poster presentation, we will discuss the history of coding theory, going in depth with cyclic and constacyclic codes, as well as discussing the many applications and current problems being resolved using algebraic coding theory.
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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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