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Asymptotically Good Codes Over Non-Abelian Groups [article]

Aria G. Sahebi, S. Sandeep Pradhan
2012 arXiv   pre-print
It has been shown that good structured codes over non-Abelian groups do exist.  ...  Specifically, we construct codes over the smallest non-Abelian group D_6 and show that the performance of these codes is superior to the performance of Abelian group codes of the same alphabet size.  ...  MAIN RESULT In this section we show the existence of good structured codes over the non-Abelian group D 6 by proving the following theorem: Theorem IV.1.  ... 
arXiv:1202.0863v2 fatcat:dpo6s5c6mzf6rpbvkvcvqntjgy

Hermitian self-dual quasi-abelian codes

Herbert S. Palines, Somphong Jitman, Romar B. Dela Cruz
2018 Journal of Algebra Combinatorics Discrete Structures and Applications  
In particular, the sub-class consisting of 1-generator quasi-abelian codes contains large families of good codes.  ...  In the case where the underlying groups are some p-groups, the actual number of resulting Hermitian self-dual quasi-abelian codes are determined. 2010 MSC: 94B15, 94B60, 16A26  ...  The asymptotic goodness of Hermitian self-dual strictly-quasi-abelian codes over F 2 2s is guaranteed by [6, Section 7] since every code over F 2 2s with generator matrix containing only elements from  ... 
doi:10.13069/jacodesmath.327399 fatcat:nqd2kdaeavhtbk27uzstwakgsy

Self-dual 2-quasi-abelian Codes [article]

Liren Lin, Yun Fan
2021 arXiv   pre-print
In a way similar to that for self-dual quasi-abelian codes of index 2, it is proved that the kind of the self-orthogonal quasi-abelian codes of index 2 is asymptotically good.  ...  A kind of self-dual quasi-abelian codes of index 2 over any finite field F is introduced.  ...  A class of codes is said to be asymptotically good if there is an asymptotic good code sequence C 1 , C 2 , · · · in the class. Let G be an abelian group of order n.  ... 
arXiv:2108.07427v1 fatcat:2yupjwvpevd6xabpjs4hru7bvy

Some randomized code constructions from group actions

L.M.J. Bazzi, S.K. Mitter
2006 IEEE Transactions on Information Theory  
Cyclic codes have been extensively studied over the last 40 years. However, it is still an open question as to whether there exist asymptotically good binary cyclic codes.  ...  We argue that by using a slightly more complex group than a cyclic group, namely the dihedral group, the existence of asymptotically good codes that are invariant under the action of the group on itself  ...  However, it was not noted that this group algebra contains asymptotically good codes.  ... 
doi:10.1109/tit.2006.876244 fatcat:nuqquj76z5ev3khaytqngmaqnu

Codes over non-Abelian groups: Point-to-point communications and computation over MAC

Aria G. Sahebi, S. Sandeep Pradhan
2012 2012 IEEE International Symposium on Information Theory Proceedings  
In this paper, we show that good structured codes over non-Abelian groups do exist.  ...  Specifically, we construct codes over the smallest non-Abelian group D6 and show that the performance of these codes is superior to the performance of Abelian group codes of the same alphabet size.  ...  Moreover, they suggest that asymptotically good group codes over non-abelian groups may not exist. This motivates a loosening of the structure of the code yet further.  ... 
doi:10.1109/isit.2012.6284269 dblp:conf/isit/SahebiP12a fatcat:yuuh4wgmnbaonbb5ofrd4lpuue

Asymptotic Properties of Quasi-Group Codes [article]

Yun Fan, Liren Lin
2022 arXiv   pre-print
The linear codes over any finite field are asymptotically good.  ...  Finally we describe the story on dihedral codes. The dihedral groups are non-abelian but near to cyclic groups (they have cyclic subgroups of index 2).  ...  In Section 6, we first prove that the quasi-abelian codes of index 2 is asymptotically good (the asymptotic goodness of quasi-abelian codes of any fixed index t ≥ 2 mentioned above can be proved in a similar  ... 
arXiv:2203.00958v1 fatcat:xly2urfegbanjb2ejsnl64wefa

Asymptotic performance of metacyclic codes [article]

Martino Borello, Pieter Moree, Patrick Solé
2019 arXiv   pre-print
In this paper, we prove that metacyclic codes form an asymptotically good family of codes.  ...  A code over a finite field F is a metacyclic code if it is a left ideal in the group algebra FG for G a metacyclic group.  ...  This would allow to show, by the combinatorial equivalence derived in [17] , that abelian group codes are asymptotically good.  ... 
arXiv:1906.07446v1 fatcat:4mbrjzhcvbe5bbbwfa4ss5ow34

Quasi-cyclic Codes of Index 1.5 [article]

Yun Fan, Hualu Liu
2015 arXiv   pre-print
We introduce quasi-cyclic codes of index 1.5, construct such codes in terms of polynomials and matrices; and prove that the quasi-cyclic codes of index 1.5 are asymptotically good.  ...  It was also shown in [5] that the quasi-abelian codes with index going to infinity are asymptotically good. The dihedral groups are the non-abelian finite groups which are nearest to cyclic groups.  ...  Bazzi and Mitter [1] obtained by a random method two asymptotically good classes of codes: (1) binary quasi-abelian codes of index 2; (2) binary dihedral group codes.  ... 
arXiv:1505.02252v1 fatcat:xdatm4fawfc3xokbxmhabqbeji

Abelian Varieties over the Field of the 20th Roots of Unity That Have Good Reduction Everywhere [chapter]

R. Schoof
2001 Applications of Algebraic Geometry to Coding Theory, Physics and Computation  
Abelian varieties over the field of the 20th roots of unity that have good reduction everywhere Abstract.  ...  We show, under assumption of GRH, that every abelian variety over Q(ζ 20 ) with good reduction everywhere is isogenous to E g for some g ≥ 0.  ...  For any abelian variety A over F = Q(ζ 20 ) with good reduction everywhere, the group scheme A[2 n ] is a 2-group scheme over O F = Z[ζ 20 ]. Proposition 3.1.  ... 
doi:10.1007/978-94-010-1011-5_15 fatcat:wbhdvwukfnfrtd4e7n4unqcokq

Thresholds of Random Quasi-Abelian Codes [article]

Yun Fan, Liren Lin
2013 arXiv   pre-print
As a consequence, there exist many asymptotically good quasi-abelian codes with any parameters attaining the GV-bound.  ...  For a random quasi-abelian code of rate r, it is shown that the GV-bound is a threshold point: if r is less than the GV-bound at δ, then the probability of the relative distance of the random code being  ...  Soon after, with the similar random method Martínez-Pérez and Willems [14] proved that self-dual doubly-even binary dihedral group codes are asymptotically good.  ... 
arXiv:1306.5377v2 fatcat:h5iebgcap5akzf74y5amdx2keq

Page 3845 of Mathematical Reviews Vol. , Issue 2002F [page]

2002 Mathematical Reviews  
The /-invariants of the corresponding Q-curves provide abelian number fields whose Galois groups over Q generically coincide with the full Atkin-Lehner group.  ...  with T and deg(N;, ) tends to oo, yields a series of asymptotically optimal curves over F,2.  ... 

On checkable codes in group algebras [article]

Martino Borello, Javier de la Cruz, Wolfgang Willems
2021 arXiv   pre-print
This means in the language of coding theory that we classify code-checkable group algebras KG which have been considered so far only for abelian groups G.  ...  Optimality of checkable codes and asymptotic results are discussed.  ...  Their results in Section 4 show that the class of group codes over these groups is asymptotically good.  ... 
arXiv:1901.10979v2 fatcat:blixwcdd4ffkffpxrisbpnvx4e

Abelian Group Codes for Channel Coding and Source Coding

Aria Ghasemian Sahebi, S. Sandeep Pradhan
2015 IEEE Transactions on Information Theory  
In this paper, we study the asymptotic performance of Abelian group codes for the the channel coding problem for arbitrary discrete (finite alphabet) memoryless channels as well as the lossy source coding  ...  Due to the non-symmetric nature of the sources and channels considered, our analysis uses a synergy of information-theoretic and group-theoretic tools.  ...  The motivation for studying Abelian group codes beyond the non-existence of finite fields over arbitrary alphabets is the following.  ... 
doi:10.1109/tit.2015.2407874 fatcat:qa3fwehqsjaxlfd4il2p4cbjze

The distribution of sandpile groups of random graphs [article]

Melanie Matchett Wood
2014 arXiv   pre-print
Our proof involves first finding the expected number of surjections from the sandpile group to any finite abelian group (the "moments" of a random variable valued in finite abelian groups).  ...  Since any particular group appears with asymptotic probability 0 (as we show), it is natural ask for the asymptotic distribution of Sylow p-subgroups of sandpile groups.  ...  In particular, this lets us see that any particular group appears asymptotically with probability 0. Corollary 9.3. Let G be a finite abelian group.  ... 
arXiv:1402.5149v2 fatcat:cg2echhnuzcpdm74uf4r5wzwy4

Group codes over fields are asymptotically good [article]

Martino Borello, Wolfgang Willems
2020 arXiv   pre-print
Following ideas of Bazzi and Mitter on group codes over the binary field, we prove that group codes over finite fields of any characteristic are asymptotically good.  ...  Group codes are right or left ideals in a group algebra of a finite group over a finite field.  ...  Group codes over fields are asymptotically good in any characteristic.  ... 
arXiv:1904.10885v2 fatcat:gbmz3xe2yzgjlmu6wgtnlqloja
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