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Cyclic codes over R k

Authors:
Steven T. Dougherty at University of Scranton
Steven T. Dougherty
Suat Karadeniz at Texas A&M University – Texarkana
Suat Karadeniz
Bahattin Yildiz at Northern Arizona University
Bahattin Yildiz
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Abstract

Cyclic codes over an infinite family of rings are defined. The general properties of cyclic codes over these rings are studied, in particular nontrivial one-generator cyclic codes are characterized. It is also proved that the binary images of cyclic codes over these rings under the natural Gray map are binary quasi-cyclic codes of index 2k . Further, several optimal or near optimal binary codes are obtained from cyclic codes over R k via this map.
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The Cyclic Codes over Ak
Yasemin Cengellenmis, Trakya University, Turkey
Steven Dougherty, University of Scranton, USA
ACCT
Pomorie,Bulgaria
June 15-21, 2012
Table of Contents
◮History
◮The Ring Akand the Gray Map on Ak
◮The Cyclic Codes over Ak
◮The Gray Image of the Self Dual Cylic Codes Over Ak
◮References
History
◮Fq: finite field
◮π:Fn
q→Fq[x]/ < xn−1>
◮C⊂Fn
q⇐⇒ π(C)⊂Fq[x]/ < xn−1>
◮A subset Cof Fn
qis called a cyclic code of length nif C
satisfies the following conditions:
*Cis a subspace of Fn
qand
*If every c= (c0,...,cn−1)∈Cthen
σ(c) = (cn−1,c0. . . , cn−2)∈C
(E.Prange,1957)
History
◮Z4
(1994) A.Hammons,P.V.Kumar,A.R.Calderbank, N.J.A
Calderbank,P.Sole
(1996) V.Pless,Z.Qian
(1997) V.Pless,P.Sole,Z.Qian
(2001) J.Wolfmann
(2003)T.Abualrub,Oehmke
(2006) S.Dougherty,S.Ling
(2003) T.Blackford
(2010) X.Kai,S.Zhu
◮Zpm
(1995)A.R.Calderbank, N.J.A Sloane
(2002)S.Ling,T.Blackford
(1997)P.Kanwar,S.R.Lopez Permouth
(2007)S.Dougherty,Y.H.Park
History
◮F2[u]/ < u2>, u2= 0
(1998) A.Bonnecaze,P.Udaya
◮Galois Ring
(1999) Wan
(2008) H.M.Kiah,K.H.Leung,S.Ling
(2009)R.Sobrani,M.Esmaeili
◮Finite Chain Ring
(2000)G.Norton,A.Salagean
(2004)H.Dihn,S.Lopez-Permouth
(2008)J.Qian,W.Ma,X.Wang
◮Fp+...+uk−1Fp
(2004)J.F.Qian,L.N.Zhang,Z.X.Zhu
(2011)M.Han,Y.Ye,S.Zhu,C.Xu,B.Dou
◮Z2+uZ2,Z2+uZ2+u2Z2
(2007)T.Abualrub,I.Siap
◮F2[u]/ < u2−1>, u2= 1
(2009) I.Siap,T.Abualrub
History
◮Z2+...+uk−1Z2
(2010)M.Al-Ashker,M.Hammoudeh
◮F2[v]/ < v2−v>, v2=v
(2010)S.Zhu,Y.Wang,M.Shi
◮F2[u,v]/ < u2,v2,uv −vu >, u2= 0,v2= 0,uv =vu
(2010)B.Yildiz,S.Karadeniz
◮F2[u1,u2,...,un]/ < u2
i,uiuj−ujui>, u2
i= 0,uiuj=ujui
(2011)S.Dougherty,B.Yildiz,S.Karadeniz
◮Fq+uFq+vFq+uvFq
(2011)X.U.Xiaofang,L.Xiusheng
◮Formal Power Series
(2011) S.Dougherty,L.Hongwei
◮M2(F2)
(2012)A.Alamadhi,H.Sboui,P.Sole,O.Yemen
History
(2010)S.Zhu,Y.Wang,M.Shi
◮R=F2[v]/ < v2−v>, v2=v
◮π:R→F2
2
(a+vb)7→ (a,a+b)
◮C1={x∈Fn
2|x,y∈Fn
2|x+vy ∈C}
C2={x+y∈Fn
2|y∈Fn
2|x+vy ∈C}
◮C= (1 + v)C1⊕vC2
Ccyclic ⇐⇒ ?C1,C2
C⊥= ?
Ccyclic ⇒C=? g1(x),g2(x)
Theorem
For any cyclic code C over R, there is a unique polynomial g (x)
such that C =<g(x)>and g (x)|xn−1where
g(x) = (1 + v)g1(x) + g2(x).
◮Ccyclic ⇒C⊥= ?
The Ring Akand the Gray Map on Ak
For integers k≥1, Ak=F2[v1,v2,...,vk]/hv2
i−vi,vivj−vjvii
where v2
i=vi,vivj=vjvi,i= 1,...,k,j= 1,...,k.
Ak={PB∈PkαBvB|αB∈F2,vB=Qi∈Bvi,B⊆ {1,2,...,k}}
The Ring Akand the Gray Map on Ak
For integers k≥1, Ak=F2[v1,v2,...,vk]/hv2
i−vi,vivj−vjvii
where v2
i=vi,vivj=vjvi,i= 1,...,k,j= 1,...,k.
Ak={PB∈PkαBvB|αB∈F2,vB=Qi∈Bvi,B⊆ {1,2,...,k}}
Example
For k = 1, A1=F2[v1]/hv2
1−v1iwhere v2
1=v1.
For k = 2, A2=F2[v1,v2]/hv2
1−v1,v2
2−v2,v1v2−v2v1iwhere
v2
1=v1,v2
2=v2,v1v2=v2v1.
The Ring Akand the Gray Map on Ak
For integers k≥1, Ak=F2[v1,v2,...,vk]/hv2
i−vi,vivj−vjvii
where v2
i=vi,vivj=vjvi,i= 1,...,k,j= 1,...,k.
Ak={PB∈PkαBvB|αB∈F2,vB=Qi∈Bvi,B⊆ {1,2,...,k}}
Example
For k = 1, A1=F2[v1]/hv2
1−v1iwhere v2
1=v1.
For k = 2, A2=F2[v1,v2]/hv2
1−v1,v2
2−v2,v1v2−v2v1iwhere
v2
1=v1,v2
2=v2,v1v2=v2v1.
Lemma
The ring Akhas characteristic 2 and cardinality 22k.
Lemma
The only unit in the ring Akis 1.
The Ring Akand the Gray Map on Ak
Theorem
The ideal hw1,w2,...,wki, where wi∈ {vi,1 + vi}for each
i= 1,2,...,k, is a maximal ideal of cardinality 22k−1.
Lemma
Let mibe a maximal ideal as above. Then there are 2ksuch ideals
and me
i=mifor all i and e ≥1.
Theorem
The ring Akis isomorphic via the Chinese Remainder Theorem to
F2k
2.Consequently, the ring Akis a principal ideal ring.
The Ring Akand the Gray Map on Ak
◮φ1:A1→F2
2
a+bv17→ φ1(a+bv1) = (a,a+b)
◮For k≥2,
φk:Ak→A2
k−1
α+βvk7→ φk(α+βvk) = (α, α +β)
◮Φk:Ak→F2k
2
Φk(γ) = φ1(φ2(...(φk−2(φk−1(φk(γ))))
The Linear Codes over Ak
A linear code Cover Akof length n is a submodule of An
k
The Linear Codes over Ak
A linear code Cover Akof length n is a submodule of An
k
◮C= (m1)C1⊕...⊕(m2k)C2k
◮C⊥= (m1)C⊥
1⊕...⊕(m2k)C⊥
2k
The Cyclic codes over Ak
Definition
A subset C of An
kis called a cyclic code of length n if C satisfies
the following conditions:
* C is a submodule of An
k
* If every c = (c0,...,cn−1)∈C then
σ(c) = (cn−1,c0. . . , cn−2)∈C
The Cyclic codes over Ak
Definition
A subset C of An
kis called a cyclic code of length n if C satisfies
the following conditions:
* C is a submodule of An
k
* If every c = (c0,...,cn−1)∈C then
σ(c) = (cn−1,c0. . . , cn−2)∈C
Theorem
Let C be a code over Akand let Cibe the binary codes given
before. The code C is cyclic if and only if Ciis a cyclic code for all
i.
The Cyclic codes over Ak
Definition
A subset C of An
kis called a cyclic code of length n if C satisfies
the following conditions:
* C is a submodule of An
k
* If every c = (c0,...,cn−1)∈C then
σ(c) = (cn−1,c0. . . , cn−2)∈C
Theorem
Let C be a code over Akand let Cibe the binary codes given
before. The code C is cyclic if and only if Ciis a cyclic code for all
i.
Corollary
If a code C over Akis cyclic then C ⊥is cyclic.
The Cyclic codes over Ak
C= (m1)C1⊕...⊕(m2k)C2k
The Cyclic codes over Ak
C= (m1)C1⊕...⊕(m2k)C2k
Theorem
Let C be a cyclic code over Akthen there exist a polynomial g (x)
in Ak[x]that divides xn−1that generates the code.
The Cyclic codes over Ak
For a polynomial, p(x) = a0+...+akxkdefine
p(x)=ak+ak−1x+...+a0xk
The Cyclic codes over Ak
For a polynomial, p(x) = a0+...+akxkdefine
p(x)=ak+ak−1x+...+a0xk
Lemma
If C is a cyclic code over Akgenerated by g (x)then C⊥is a cyclic
code generated by (xn−1/g(x)).
The Gray image of the Self Dual Cyclic Codes over Ak
Definition
Let a∈F2kn
2with a= (a0,...,a2kn−1) = (a(0)|a(1) |...|a(2k−1)),
a(i)∈Fn
2for i = 0,1,...,2k−1.Let σ⊗2kbe the map from F2kn
2
to F2kn
2given by σ⊗2k(a) = (σ(a(0))|. . . |σ(a(2k−1))) where σis the
usual shift (c0,...,cn−1)7→ (cn−1,c0, . . . , cn−2)on Fn
2.
A code C of length 2kn over F2is said to be quasi-cyclic of index
2kif σ⊗2k(C) = C .
The Gray image of the Self Dual Cyclic Codes over Ak
Definition
Let a∈F2kn
2with a= (a0,...,a2kn−1) = (a(0)|a(1) |...|a(2k−1)),
a(i)∈Fn
2for i = 0,1,...,2k−1.Let σ⊗2kbe the map from F2kn
2
to F2kn
2given by σ⊗2k(a) = (σ(a(0))|. . . |σ(a(2k−1))) where σis the
usual shift (c0,...,cn−1)7→ (cn−1,c0, . . . , cn−2)on Fn
2.
A code C of length 2kn over F2is said to be quasi-cyclic of index
2kif σ⊗2k(C) = C .
Corollary
The image of a cyclic self dual code of length n over Akis a length
2kn self dual quasi-cyclic code of index 2k.
Bibliography
◮C. Bachoc,Application of Coding Theory to the Construction
of Modular Lattices,J.Combin Theory Ser.A78,92-119,1997
◮Y.Cengellenmis,A.Dertli,S.Dougherty,Cyclic and Skew Cyclic
Codes over an Infinite Family of Rings with Gray Map,in
submission
◮Y.Cengellenmis,On the Cylic codes over
F3+vF3,International Journal of Algebra,4,no 6,253-259,2010
◮N.J.A Sloane,J.G.Thompson, Cyclic Self Dual Codes,IEEE
Trans.Information Theory,IT-29,364-366,1983
◮S.Zhu,Y.Wang,M.J.Shi,Cyclic Codes over
F2+vF2,ISIT,Korea,2009
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In this work, we investigate linear codes over the ring F2+uF2+vF2+uvF2{\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2} . We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary to the rings that have hitherto been studied in coding theory. Lee weights and Gray maps for these codes are defined by extending on those introduced in works such as Betsumiya et al. (Discret Math 275:43–65, 2004) and Dougherty et al. (IEEE Trans Inf 45:32–45, 1999). We then characterize the F2+uF2+vF2+uvF2{\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2} -linearity of binary codes under the Gray map and give a main class of binary codes as an example of F2+uF2+vF2+uvF2{\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2} -linear codes. The duals and the complete weight enumerators for F2+uF2+vF2+uvF2{\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2} -linear codes are also defined after which MacWilliams-like identities for complete and Lee weight enumerators as well as for the ideal decompositions of linear codes over F2+uF2+vF2+uvF2{\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2} are obtained.
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On the Structure of Linear and Cyclic Codes Over a Finite Chain Ring.
Article
  • Jul 2000
We generalise structure theorems of Calderbank and Sloane for linear and cyclic codes over ℤ pa to a finite chain ring. Our results are more detailed and do not use non-trivial results from Commutative Algebra.
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