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

2000
*
arXiv
*
pre-print

We also provide examples

arXiv:quant-ph/0008129v1
fatcat:4unxtcjttzb67ojixte3tge544
*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*). ...##
###
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]

2022
*
arXiv
*
pre-print

Secondly, a necessary and sufficient condition for a class

arXiv:2206.13810v1
fatcat:nfp7hpejsjhrngxknwitdhdbku
*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*. ...##
###
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

2007
*
2007 IEEE Information Theory Workshop
*

Specifically, limits

doi:10.1109/itw.2007.4313064
fatcat:znh4zgn3r5cx7ez7yvilnvylvm
*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. ...##
###
Quasi-Cyclic Complementary Dual Code
[article]

2015
*
arXiv
*
pre-print

LCD

arXiv:1506.01971v1
fatcat:5ywoz2w6tffrfnxlmx3dimed6m
*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. ...##
###
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™). ...

##
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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 ...##
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The q-ary image of some q/sup m/-ary cyclic codes: permutation group and soft-decision decoding

2002
*
IEEE Transactions on Information Theory
*

Using a particular construction

doi:10.1109/tit.2002.1013146
fatcat:iew3vhmygfg2volkqlaamcloa4
*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. ...##
###
New Quantum and LCD Codes over Finite Fields of Even Characteristic

2021
*
Defence Science Journal
*

Also, we characterise LCD

doi:10.14429/dsj.71.16641
fatcat:einuru6uovdc3gl6gjis2dvnuu
*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. ...##
###
Skew Cyclic codes over _q+u_q+v_q+uv_q
[article]

2015
*
arXiv
*
pre-print

We investigate the structural properties

arXiv:1504.07831v1
fatcat:33fvcqim7bag5ilhxymhurwtwu
*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. ...##
###
Cyclic additive codes

2012
*
Journal of Algebra
*

A

doi:10.1016/j.jalgebra.2012.08.031
fatcat:2xmqyxsvx5bqxdkomeozwpchtu
*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. ...##
###
Quasi-cyclic codes over Z/sub 4/ and some new binary codes

2002
*
IEEE Transactions on Information Theory
*

Among the new

doi:10.1109/tit.2002.1013145
fatcat:u5bxq4wzmzg7njvx2bymqzv2ey
*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. ...##
###
Skew cyclic codes over $F_{q}+uF_{q}+vF_{q}+uvF_{q}$

2015
*
Journal of Algebra Combinatorics Discrete Structures and Applications
*

We investigate the structural properties

doi:10.13069/jacodesmath.90080
fatcat:dmcljs5iuneiphes24ieymyvfy
*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. ...##
###
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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