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Quantum codes from cyclic codes over GF(4^m)
[article]
2000
arXiv
pre-print
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We also provide examples of these codes some of which meet the known bounds for quantum codes. ...
We provide a construction for quantum codes (hermitian-self-orthogonal codes over GF(4)) starting from cyclic codes over GF(4^m). ...
The q-ary image of C with respect to B is defined to be d B (C) which is a linear code of length mn over GF (q). ...
arXiv:quant-ph/0008129v1
fatcat:4unxtcjttzb67ojixte3tge544
Page 5700 of Mathematical Reviews Vol. , Issue 2003g
[page]
2003
Mathematical Reviews
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Summary: “Using a particular construction of generator matri- ces of the g-ary image of g’’-ary cyclic codes, it is proved that some of these codes are invariant under the action of particu- lar permutation ...
Of course, these groups are closely related to the general linear group GL(n,q), the projective linear group PI'L(n,q) and the wreath product Sym(g —1)?PIL(n,q). ...
Gray Images of Cyclic Codes over ℤ_p^2 and ℤ_pℤ_p^2
[article]
2022
arXiv
pre-print
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Secondly, a necessary and sufficient condition for a class of ℤ_pℤ_p^2-additive codes whose Gray images are linear (not necessarily cyclic) over ℤ_p is given. ...
Moreover, as for some special families of cyclic codes over ℤ_9 and ℤ_3 ℤ_9, the linearity of the Gray images is determined. ...
We determined some special Z p Z p 2 -additive cyclic codes with gcd(β, p) = 1 whose Gray images are linear p-ary codes. ...
arXiv:2206.13810v1
fatcat:nfp7hpejsjhrngxknwitdhdbku
Page 5762 of Mathematical Reviews Vol. , Issue 96i
[page]
1996
Mathematical Reviews
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a q”’-ary cyclic code. ...
Then d, is an F,-linear map and
aj = Li=0
4 (za(e)) = z'd,(a(z)). If V is a q”-ary [n, k] cyclic code, its q- ary image with respect to the basis a is d,(V), where d,(V) = {d,(a(z)): a(z) € V}. ...
The Complexity Limits of Graphical Models for Linear Codes
2007
2007 IEEE Information Theory Workshop
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Specifically, limits on the space of graphical models for a given code are examined. A number of results concerning these limits (e.g., the cut-set bound, the square-root bound) are first reviewed. ...
Since the discovery of turbo codes in the 1990s, much of the work concerning graphical models for codes has focused on constructing models in order to define a new code. ...
Specifically, let G C be a q m -ary graphical model for the linear code C over F q defined on an index set I. ...
doi:10.1109/itw.2007.4313064
fatcat:znh4zgn3r5cx7ez7yvilnvylvm
Quasi-Cyclic Complementary Dual Code
[article]
2015
arXiv
pre-print
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LCD codes are linear codes that intersect with their dual trivially. Quasi cyclic codes that are LCD are characterized and studied by using their concatenated structure. ...
Some asymptotic results are derived. Hermitian LCD codes are introduced to that end and their cyclic subclass is characterized. Constructions of QCCD codes from codes over larger alphabets are given. ...
Note that if C is a cyclic code of length m over F q ℓ , then ϕ(C) is a linear code over F q of length mℓ, which is closed under shift of codewords by ℓ units. Example 5.4. ...
arXiv:1506.01971v1
fatcat:5ywoz2w6tffrfnxlmx3dimed6m
Page 3223 of Mathematical Reviews Vol. , Issue 96e
[page]
1996
Mathematical Reviews
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On cyclic codes which are g-ary images of linear codes. (English summary)
Appl. Algebra Engrg. Comm. Comput. 2 (1992), no. 3, 163-170. ...
In this paper the author considers (mn,mk) cyclic codes over GF(q) which are q-ary images of (n,k) cyclic codes over GF(q™). ...
Page 3759 of Mathematical Reviews Vol. , Issue 2002E
[page]
2002
Mathematical Reviews
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The underlying codes are cyclic codes.”
2002e:94128 94B15 11T71
Mouaha, Christophe; Schiffels, Gerhard (D-BLFM; Bielefeld)
All q’"-ary cyclic codes with cyclic g-ary image are known. ...
This paper complements these earlier papers, gives a full answer to the characterization problem, and shows that there are no more g”-ary cyclic codes with cyclic q-ary image than those given by Séguin ...
The q-ary image of some q/sup m/-ary cyclic codes: permutation group and soft-decision decoding
2002
IEEE Transactions on Information Theory
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Using a particular construction of generator matrices of the -ary image of -ary cyclic codes, it is proved that some of these codes are invariant under the action of particular permutation groups. ...
Index Terms-Permutation groups, -ary image of -ary cyclic codes, soft-decision decoding. ...
Koetter for his helpful comments on the drafting of this correspondence. ...
doi:10.1109/tit.2002.1013146
fatcat:iew3vhmygfg2volkqlaamcloa4
New Quantum and LCD Codes over Finite Fields of Even Characteristic
2021
Defence Science Journal
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Also, we characterise LCD codes of length l with respect to their generator polynomials and prove that F2m − image of an LCD code of length l is an LCD code of length 2l . ...
Finally, we provide several optimal LCD codes from the Gray images of LCD codes over F2m + uF2m . ...
In this context, the most used linear codes are cyclic codes which efficiently help to pursue. ...
doi:10.14429/dsj.71.16641
fatcat:einuru6uovdc3gl6gjis2dvnuu
Skew Cyclic codes over _q+u_q+v_q+uv_q
[article]
2015
arXiv
pre-print
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We investigate the structural properties of skew cyclic codes over R through a decomposition theorem. Furthermore, we give a formula for the number of skew cyclic codes of length n over R. ...
In this paper, we study skew cyclic codes over the ring R=_q+u_q+v_q+uv_q, where u^2=u,v^2=v,uv=vu, q=p^m and p is an odd prime. ...
First, we introduce the structure of linear codes over R and show the structural properties of skew cyclic codes over R. ...
arXiv:1504.07831v1
fatcat:33fvcqim7bag5ilhxymhurwtwu
Cyclic additive codes
2012
Journal of Algebra
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A q-linear q m -ary code C is cyclic in the monomial sense if it is invariant under an element g( A 0 , . . . , A n−1 , π) of the monomial group where π ∈ S n is an n-cycle. ...
We develop the theory of additive cyclic codes. Definition 4. A code C of length n is cyclic in the permutation sense if there is an n-cycle π ∈ S n such that π(C) = C. ...
Cyclic additive codes in the monomial sense Let C be a q m -ary q-linear code of length n, which is cyclic in the monomial sense. ...
doi:10.1016/j.jalgebra.2012.08.031
fatcat:2xmqyxsvx5bqxdkomeozwpchtu
Quasi-cyclic codes over Z/sub 4/ and some new binary codes
2002
IEEE Transactions on Information Theory
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Among the new codes, the lift of the famous Golay code to produces a new binary code, a (92 2 28)-code, which is the best among all binary codes (linear or nonlinear). ...
Recently, (linear) codes over and quasi-cyclic (QC) codes (over fields) have been shown to yield useful results in coding theory. ...
Ferrar of the Department of Mathematics, The Ohio State University, for his useful discussions and the referees for the suggestions. ...
doi:10.1109/tit.2002.1013145
fatcat:u5bxq4wzmzg7njvx2bymqzv2ey
Skew cyclic codes over $F_{q}+uF_{q}+vF_{q}+uvF_{q}$
2015
Journal of Algebra Combinatorics Discrete Structures and Applications
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We investigate the structural properties of skew cyclic codes over R through a decomposition theorem. Furthermore, we give a formula for the number of skew cyclic codes of length n over R. ...
In this paper, we study skew cyclic codes over the ring R = Fq + uFq + vFq + uvFq, where u 2 = u, v 2 = v, uv = vu, q = p m and p is an odd prime. ...
In the following, we assume G i are generator matrices of 9-ary linear codes C i for i = 1, 2, 3, 4, respectively. ...
doi:10.13069/jacodesmath.90080
fatcat:dmcljs5iuneiphes24ieymyvfy
Page 2050 of Mathematical Reviews Vol. , Issue 98C
[page]
1998
Mathematical Reviews
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Summary: “For (n,q) = p’, where p=ch(F,), s >1, and V is a q”-ary repeated-root cyclic code of length n with generator polynomial g(x), we give a partial answer about whether the g- ary image of V is cyclic ...
Summary: “The extended cyclic codes of length p”, p a prime, which are invariant under the affine group acting on Fp», are called affine-invariant codes. ...
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