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On weights in duadic abelian codes

Qiaoliang Li
2003 Discrete Mathematics  
In this note, we prove that if C is a duadic binary abelian code with splitting = −1 and the minimum odd weight of C satisÿes d 2 − d + 1 = n, then d(d − 1) ¿ n + 11.  ...  A series of open problems on this subject are proposed.  ...  The author are grateful to the referees for many helpful comments on an early version of this paper.  ... 
doi:10.1016/s0012-365x(02)00673-8 fatcat:33uqhdl6tjbk3dtox62m7mdk3e

Duadic codes over Z/sub 2k/

San Ling, P. Sole
2001 IEEE Transactions on Information Theory  
Duadic codes constitute a well-known family of binary cyclic codes. They are generalized in this correspondence to the setting of Abelian codes over the ring .  ...  They are also grateful to the anonymous referees and the Associate Editor for very helpful comments on the manuscript. This work was performed when S.  ...  Finally, in Section VI, we define duadic codes and generalized duadic codes over Z Z Z 2k and prove some results on self-duality, isoduality, and Type II property of Abelian Z Z Z 2k codes.  ... 
doi:10.1109/18.923740 fatcat:ptda5uzujvftvafah35lcs244u

Duadic Codes over F 2 + u F 2

San Ling, Patrick Solé
2001 Applicable Algebra in Engineering, Communication and Computing  
Duadic codes over F 2 +uF 2 are introduced as abelian codes by their zeros. This is the function field analogue of duadic codes over Z 4 introduced recently by Langevin and Solé.  ...  We classify them in modest lengths and exhibit interesting non-cyclic examples.  ...  Isoduality For an abelian code C in R[G], a multiplier α acts on C by permutation of the coordinates.  ... 
doi:10.1007/s002000100079 fatcat:btccoloy4vhgtdoy76vkfcarvi

Duadic codes over $ \mathbb{Z}_4+u\mathbb{Z}_4 $

Raj Kumar, Maheshanand Bhaintwal
2021 Advances in Mathematics of Communications  
Some conditions on the existence of self-dual augmented and extended duadic codes over Z 4 + uZ 4 are determined.  ...  We study these codes by considering them as a class of abelian codes. Some results related to self-duality and self-orthogonality of duadic codes are presented.  ...  have studied constructions of one-Gray weight codes and two-Gray weight codes over Z 4 + uZ 4 based on a Gray map from Z 4 + uZ 4 to Z 2 4 . Gao et al.  ... 
doi:10.3934/amc.2020135 fatcat:xvizecdcirb2rg34tir3g4r2va

Duadic Z4-Codes

Philippe Langevin, Patrick Solé
2000 Finite Fields and Their Applications  
An arithmetic test on the length for the existence of nontrivial abelian self-dual codes is derived.  ...  A natural generalization of both the supplemented quadratic residue codes and the binary duadic codes is introduced.  ...  Noncyclic abelian codes of length (45 are not self-dual but they can be (like duadic codes in the binary world) isodual (i.e., isometric to their dual).  ... 
doi:10.1006/ffta.2000.0285 fatcat:shzglpcqqnb3bh75kfp3a42sku

Split group codes

Cunsheng Ding, D.R. Kohel
2000 IEEE Transactions on Information Theory  
The construction, using the group algebra of an Abelian group and a special partition or splitting of the group, yields quadratic residue codes, duadic codes, and their generalizations as special cases  ...  residue codes.  ...  Since most of the features of duadic codes also carry over to the Abelian group algebras, Rushanan [16] defines duadic codes in this setting, but reverts to a nonconstructive idempotent definition.  ... 
doi:10.1109/18.825811 fatcat:euahl2qfqbb7lar5smgmpqxfmi

Duadic codes and difference sets

Joseph J. Rushanan
1991 Journal of combinatorial theory. Series A  
Then we prove the main theorem, which gives sufficient conditions for the code of a difference set to be embedded in a duadic code (over an appropriate field).  ...  Throughout, G is an abelian group of order u (written additively) and [F = GF( q), where we assume that gcd(u, q) = 1. All of our calculations take place in the group ring R = lF [G].  ...  DUADIC CODES Duadic codes are abelian group codes defined by a pair of idempotents. Duadic codes were first studied for GF (2) and GF(4) by Pless et al.  ... 
doi:10.1016/0097-3165(91)90049-m fatcat:aaq2a3btyfg4dcexvjobi5n4te

Existence of abelian group code partitions

Harold N Ward, L Zhu
1994 Journal of combinatorial theory. Series A  
This note presents necessary and sufficient conditions for the existence of generalizations of m-adic residue codes in Abelian group algebras.  ...  In this note we shall deal with a generalization of duadic codes for Abelian groups analogous to the cyclic m-adic residue codes in [2] .  ...  Brualdi and Pless generalized them to polyadic cyclic codes [2] and Rushanan did so to duadic Abelian group codes [9] .  ... 
doi:10.1016/0097-3165(94)90017-5 fatcat:ukjukqubhbgbrilo7bqfyjpmei

Duadic Group Algebra Codes [article]

Salah A. Aly, Andreas Klappenecker, Pradeep Kiran Sarvepalli
2007 arXiv   pre-print
Duadic group algebra codes are a generalization of quadratic residue codes. This paper settles an open problem raised by Zhu concerning the existence of duadic group algebra codes.  ...  These codes can be used to construct degenerate quantum stabilizer codes that have the nice feature that many errors of small weight do not need error correction; this fact is illustrated by an example  ...  Odd-like Weights in Duadic Group Algebra Codes Our first result is a slight generalization of a theorem by Zhu [14] .  ... 
arXiv:cs/0701060v1 fatcat:rtdr46lxanhstkxxfoxmvsskjq

Page 6760 of Mathematical Reviews Vol. , Issue 2000i [page]

2000 Mathematical Reviews  
The authors extend the classical construction of Reed-Muller (RM) codes via elementary abelian 2-groups to an arbitrary finite abelian group and define a family of RM-type codes.  ...  In this case it contains j, the all-ones vector. For a given code C define the code D = C+ + F)j. Some properties of these two codes associated with an r-net are established.  ... 

Page 3151 of Mathematical Reviews Vol. , Issue 2003d [page]

2003 Mathematical Reviews  
We also give a method for the determination of the dimension of the codes and finally some results on the weight hierarchy are presented.”  ...  We highlight some of the recent developments in the study of duadic codes.”  ... 

Page 1905 of Mathematical Reviews Vol. , Issue 95c [page]

1995 Mathematical Reviews  
On the one hand duadic codes are generalized to triadic and polyadic codes, and on the other hand to codes generated by difference sets in abelian groups.  ...  Duadic codes are generalizations of quadratic residue codes. The paper gives a survey of recent results on duadic codes and their generalizations in various directions.  ... 

Page 5929 of Mathematical Reviews Vol. , Issue 97I [page]

1997 Mathematical Reviews  
The author proves a square-root bound on the weights of odd-like words in a duadic group algebra code. When the splitting involved is the inverse map g — g~', the bound on the weight d is that d?  ...  A. van Tilborg [On weights in codes, Onderafdeling der Wiskunde, Technische Hogeschool Eindhoven, Eindhoven, 1971; MR 48 #10659].)  ... 

Nonlinear Analysis of the Iterative Decoding of Parallel Concatenated Convolutional Codes

F. Lehmann, G.M. Maggio
2005 IEEE Transactions on Information Theory  
A lower bound on this map is derived based upon the weight enumerator of the constituent codes, thus enabling the characterization of the dynamics of the decoder in terms of fixed points, along with the  ...  In this correspondence, we introduce a simple one-dimensional (1-D) nonlinear map to describe the iterates of the bit-error rate (BER) of parallel-concatenated convolutional codes (PCCC) on the binary-input  ...  Let A w;d represent the number of codewords in the terminated constituent code with information weight w and weight d.  ... 
doi:10.1109/tit.2005.847751 fatcat:4f24u4eliverhagdarlq7hoocm

Page 9831 of Mathematical Reviews Vol. , Issue 2003m [page]

2003 Mathematical Reviews  
(ML) decoding.” 2003m:94089 94B25 Li, Qiaoliang (PRC-HNN; Changsha) On weights in duadic abelian codes.  ...  Summary: “In this note, we prove that if C is a duadic binary abelian code with splitting « = u_; and the minimum odd weight of C satisfies d? —d +14n, then d(d —1) >n+11.  ... 
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