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Quantum codes from cyclic codes over GF(4^m) [article]

Andrew Thangaraj, Steven McLaughlin
2000 arXiv   pre-print
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  
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]

Minjia Shi, Xuan Wang
2022 arXiv   pre-print
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  
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

Thomas R. Halford, Keith M. Chugg
2007 2007 IEEE Information Theory Workshop  
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]

Cem Güneri̇, Buket Özkaya, Patrick Solé
2015 arXiv   pre-print
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  
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  
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

J. Lacan, E. Delpeyroux
2002 IEEE Transactions on Information Theory  
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

Habibul Islam, Om Prakash
2021 Defence Science Journal  
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]

Ting Yao, Minjia Shi, Patrick Solé
2015 arXiv   pre-print
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

Jürgen Bierbrauer
2012 Journal of Algebra  
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

N. Aydin, D.K. Ray-Chaudhuri
2002 IEEE Transactions on Information Theory  
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 $F_{q}+uF_{q}+vF_{q}+uvF_{q}$

Ting Yao, Minjia Shi, Patrick Solé
2015 Journal of Algebra Combinatorics Discrete Structures and Applications  
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  
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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