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(1+2u)-constacyclic codes over
Abstract
Let where denotes the ring of
integers modulo 4 and . In the present paper, we introduce a new Gray
map from to We study (1+2u)-constacyclic codes
over R of odd lengths with the help of cyclic codes over R. It is proved
that the Gray image of (1+2u)-constacyclic codes of length n over R are
cyclic codes of length 2n over . Further, a number of linear
codes over as the images of (1+2u)-constacyclic codes over R
are obtained.
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For any prime p, all constacyclic codes of length over the ring are considered. The units of the ring are of the forms γ and , where , and γ are nonzero elements of , which provides such constacyclic codes. First, the structure and Hamming distances of all constacyclic codes of length over the finite field are obtained; they are used as a tool to establish the structure and Hamming distances of all -constacyclic codes of length over . We then classify all cyclic codes of length over and obtain the number of codewords in each of those cyclic codes. Finally, a one-to-one correspondence between cyclic and γ-constacyclic codes of length over is constructed via ring isomorphism, which carries over the results regarding cyclic codes corresponding to γ-constacyclic codes of length over .
This paper studies (1 + u)-constacyclic codes over the ring F 2 + uF 2 + vF 2 + uvF 2. It is proved that the image of a (1+u)-constacyclic code of length n over F 2+uF 2+vF 2+uvF 2 under a Gray map is a distance invariant binary quasi-cyclic code of index 2 and length 4n. A set of generators of such constacyclic codes for an arbitrary length is determined. Some optimal binary codes are obtained directly from (1 + u)-constacyclic codes over F 2 + uF 2 + vF 2 + uvF 2. © 2012 Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg.
Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson, Kerdock, Preparata, Goethals, and Delsarte-Goethals. It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z_4, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the Z_4 domain implies that the binary images have dual weight distributions. The Kerdock and "Preparata" codes are duals over Z_4 -- and the Nordstrom-Robinson code is self-dual -- which explains why their weight distributions are dual to each other. The Kerdock and "Preparata" codes are Z_4-analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z_4, which greatly simplifies encoding and decoding. An algebraic hard-decision decoding algorithm is given for the "Preparata" code and a Hadamard-transform soft-decision decoding algorithm for the Kerdock code. Binary first- and second-order Reed-Muller codes are also linear over Z_4, but extended Hamming codes of length n >= 32 and the Golay code are not. Using Z_4-linearity, a new family of distance regular graphs are constructed on the cosets of the "Preparata" code.
This paper presents some basic theorems giving the structure of cyclic codes of lengthn over the ring of integers modulop
a
and over thep-adic numbers, wherep is a prime not dividingn. An especially interesting example is the 2-adic cyclic code of length 7 with generator polynomialX
3
+λX
2
+(λ−1)X−1, where λ satisfies λ2 - λ + 2 = 0. This is the 2-adic generalization of both the binary Hamming code and the quaternary octacode (the latter being equivalent to the Nordstrom-Robinson code). Other examples include the 2-adic Golay code of length 24 and the 3-adic Golay code of length 12.
The structures of cyclic and negacyclic codes of length n and their duals over a finite chain ring R are established when n is not divisible by the characteristic of the residue field R~. Some cases where n is divisible by the characteristic of the residue field R~ are also considered. Namely, the structure of negacyclic codes of length 2t over Z2m and that of their duals are derived.
Certain notorious nonlinear binary codes contain more codewords
than any known linear code. These include the codes constructed by
Nordstrom-Robinson (1967), Kerdock (1972), Preparata (1968), Goethals
(1974), and Delsarte-Goethals (1975). It is shown here that all these
codes can be very simply constructed as binary images under the Gray map
of linear codes over Z 4, the integers mod 4
(although this requires a slight modification of the Preparata and
Goethals codes). The construction implies that all these binary codes
are distance invariant. Duality in the Z 4 domain
implies that the binary images have dual weight distributions. The
Kerdock and “Preparata” codes are duals over
Z 4-and the Nordstrom-Robinson code is
self-dual-which explains why their weight distributions are dual to each
other. The Kerdock and “Preparata” codes are
Z 4-analogues of first-order Reed-Muller and extended
Hamming codes, respectively. All these codes are extended cyclic codes
over Z 4, which greatly simplifies encoding and
decoding. An algebraic hard-decision decoding algorithm is given for the
“Preparata” code and a Hadamard-transform soft-decision
decoding algorithm for the I(Kerdock code. Binary first- and
second-order Reed-Muller codes are also linear over Z <sub>4
</sub>, but extended Hamming codes of length n⩾32 and the Golay code
are not. Using Z 4-linearity, a new family of
distance regular graphs are constructed on the cosets of the
“Preparata” code
Constacyclic codes are an important class of linear codes in coding theory. Many optimal linear codes are directly derived from constacyclic codes. In this paper, (1 − uv)-constacyclic codes over the local ring are studied. It is proved that the image of a (1 − uv)-constacyclic code of length n over under a Gray map is a distance invariant quasi-cyclic code of index p
2 and length p
3
n over . Several examples of optimal linear codes over from (1 − uv)-constacyclic codes over are given.
We study the structure of (1+u)-constacyclic codes of an arbitrary length n over the ring F 2 +uF 2 . We find a set of generators for each (1+u)-constacyclic code and its dual. We study the rank of cyclic codes and find their minimal spanning sets. We prove that the Gray image of a (1+u)-constacyclic code is a binary cyclic code of length 2n. We conclude by giving examples of constacyclic codes and their Gray image binary codes. We give a direct construction of a [12,7,4] linear binary cyclic code that match the Hamming distance of the best binary code with length 12 and dimension 7.
Cyclic codes of odd length over Z(4) have been studied by many authors. But what is the form of cylic codes of even length? The structure of cyclic codes of length n = 2(e), for any positive integer a is considered. We show that any cyclic code is an ideal in the ring R-n = Z(4)[x]/<x(n)-1>. We show that the ring R-n is a local ring but not a principal ideal ring. Also, we find the set of generators for cyclic codes. Examples of cyclic codes of such length are given. (C) 2003 Published by Elsevier Science B.V.
We determine the structure of cyclic codes over
for arbitrary even length giving the generator polynomial for these codes. We determine the number of cyclic codes for a given length. We describe the duals of the cyclic codes, describe the form of cyclic codes that are self-dual and give the number of these codes. We end by examining specific cases of cyclic codes, giving all cyclic self-dual codes of length less than or equal to 14.
Linear codes are considered over the ring Z_4+uZ_4, a non-chain extension of
Z_4. Lee weights, Gray maps for these codes are defined and MacWilliams
identities for the complete, symmetrized and Lee weight enumerators are proved.
Two projections from Z_4+uZ_4 to the rings Z_4 and F_2+uF_2 are considered and
self-dual codes over Z_4+uZ_4 are studied in connection with these projections.
Finally three constructions are given for formally self-dual codes over
Z_4+uZ_4 and their Z_4-images together with some good examples of formally
self-dual Z_4-codes obtained through these constructions.
Given an integer m which is a product of distinct primes pi, a method is given for constructing codes over the ring of integers modulo m from cyclic codes over GF(pi). Specifically, if we are given a cyclic (n, ki) code over GF(pi) with minimum Hamming distance di, for each i, then we construct a code of block length n over the integers modulo m with πi pkii codewords, which is both linear and cyclic and has minimum Hamming distance minidi.
This paper classifies all cyclic codes over Z4 of length 2n,n odd. Descriptions are given in terms of discrete Fourier transforms, generator polynomials, parity check matrices, and the concatenated (a+b|b) construction. Some results about the minimum Lee weights of these codes and self-dual codes are also included.
Linear codes over the ring of integers modulo q = pr, p a prime, are considered. Natural analogs to Hamming, Reed—Solomon, and BCH codes over finite fields are defined and their properties investigated. Some ring theoretic problems encountered are discussed.
We determine all linear cyclic codes over ℤ4 of odd length whose Gray images are linear codes or, equivalently, whose Netchaev-Gray image are linear cyclic codes.
This paper generalizes the results from Wolfmann (see ibid., vol.45, p.2527-2532, Nov. 1999 and vol.47, p.1773-1779, July 2001), classifying all negacyclic codes over Z4 of even length using a transform approach. It is then shown which linear binary cyclic codes are images of negacyclic codes under the Gray map. In the process, the concatenated structure of both negacyclic codes and binary repeated-root cyclic codes is given.
We determine all linear cyclic codes over Z 4
of odd length whose Gray images are linear codes (or, equivalently,
whose Nechaev-Gray (1989) image are linear cyclic codes or are linear
cyclic codes)
Cyclic codes over Z 4 of length 2 e , Disc
- T Abualrub
- R Oehmke
T. Abualrub and R. Oehmke, Cyclic codes over Z 4 of length 2 e, Disc. App. Math. 128(2003), 3-9.
On cyclic codes over Z 4 + uZ 4 and their Z 4 images
- B Yildiz
- Nuh Audin
B. Yildiz and Nuh Audin, On cyclic codes over Z 4 + uZ 4 and their Z 4 images, Int.
J. Information and Coding Theory 2(2014), 226-237.
- P Kanwar
- S R Lopez-Permouth
P. Kanwar and S. R. Lopez-Permouth, Cyclic codes over the integers modulo p m,
Finite Fields Appl. 3(1997), 334-352.
- R K Bandi
- M Bhaintwal
R. K. Bandi and M. Bhaintwal, cyclic codes over Z 4 + uZ 4, arXiv: 1501.01327v1
[cs.IT](2015).
Z k+1 p -linear codes
- S Ling
- J Blackford
S. Ling and J. Blackford, Z k+1
p -linear codes, IEEE Trans. Inform. Theory 48(2002),
2592-2605.
- R K Bandi
- M Bhaintwal
R. K. Bandi and M. Bhaintwal, cyclic codes over Z 4 + uZ 4, arXiv: 1501.01327v1
[cs.IT](2015).
Linear, cyclic and constacyclic over S 4 = F 2 + uF 2 + u 2 F 2 + u 3 F 2
- Z O Ozger
- U U Kara
- B Yildiz
Z. O. Ozger, U. U. Kara and B. Yildiz, Linear, cyclic and constacyclic over S 4 =
F 2 + uF 2 + u 2 F 2 + u 3 F 2, Filomat 28(2014), 897-906.