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The Dynamics of Group Codes: Dual Abelian Group Codes and Systems

G.D. Forney, M.D. Trott
2004 IEEE Transactions on Information Theory  
Fundamental results concerning the dynamics of abelian group codes (behaviors) and their duals are developed.  ...  The dual of a complete code or system is finite, and the dual of a Laurent code or system is (anti-)Laurent.  ...  We are particularly indebted to Thomas Mittelholzer for helpful suggestions and references with regard to the issues of "closed projections" and "completeness," and to Fabio Fagnani for correction of mathematical  ... 
doi:10.1109/tit.2004.838340 fatcat:wdik7vzm45hwffmmt3nq7wvowa

Tanner graphs for group block codes and lattices: construction and complexity

A.H. Banihashemi, F.R. Kschischang
2001 IEEE Transactions on Information Theory  
We also study the problem of finding low-complexity TGs for Abelian group block codes and lattices, and derive tight lower bounds on the label-code complexity of lattices.  ...  We develop a Tanner graph (TG) construction for an Abelian group block code with arbitrary alphabets at different coordinates, an important application of which is the representation of the label code  ...  ACKNOWLEDGMENT The authors wish to acknowledge the anonymous referees for their helpful comments and suggestions, which greatly improved the presentation of the paper. They also wish to thank Dr. G.  ... 
doi:10.1109/18.910592 fatcat:a2wl44cfobcf3g2rdtq5zshnpa

Self-duality and codings for expansive automorphisms

Alex Clark, Robbert Fokkink
2007 Topology and its Applications  
Lind and Schmidt have shown that for certain ergodic Z k -actions on a compact abelian group Γ , the homoclinic group H is isomorphic to the Pontryagin dual of Γ .  ...  In this paper, we present a dynamical interpretation of R if k = 1: it is a product of the stable group and the unstable group of Γ , under a suitable topology.  ...  Acknowledgements The first author was partly supported by an NWO visitor's grant. We would like to thank Valérie Berthé, Karma Dajani, and Hendrik Lenstra for helpful conversations.  ... 
doi:10.1016/j.topol.2006.04.032 fatcat:2sozoaayxncm7pnvinkerj65pi

Codes on Graphs: Fundamentals [article]

G. David Forney Jr
2014 arXiv   pre-print
This paper develops a fundamental theory of realizations of linear and group codes on general graphs using elementary group theory, including basic group duality theory.  ...  for cycle-free codes; results showing that all constraint codes except interface nodes may be assumed to be trim and proper, and that the interesting part of a cyclic realization is its "2-core;" notions  ...  Acknowledgments I thank Heide Gluesing-Luerssen for an ongoing research collaboration under which many of these results were first developed, and for extensive comments on this paper.  ... 
arXiv:1306.6264v2 fatcat:zo6zwanszjcbvc44p36dolibk4

Self Duality and Codings for Expansive Group Automorphisms [article]

Alex Clark, Robbert Fokkink
2005 arXiv   pre-print
Lind and Schmidt have shown that the homoclinic group of a cyclic ^k algebraic dynamical system is isomorphic to the dual of the phase group.  ...  The constructions naturally lead to an almost 1-1-coding of certain Pisot automorphisms by their associated β-shift, generalizing similar results for Pisot automorphisms of the torus.  ...  If the phase space X is an abelian topological group and if the action x → g · x is an automorphism, then the system is called an algebraic dynamical system. In this case, X is a Z[G] module.  ... 
arXiv:math/0501469v1 fatcat:mqbembgyb5dqvexjlcptgeuwmm

Quantum Error-Control Codes [article]

Martianus Frederic Ezerman
2020 arXiv   pre-print
It is to appear as a book chapter in "A Concise Encyclopedia of Coding Theory," edited by C. Huffman, P. Sole and J-L Kim, to be published by CRC Press.  ...  The article surveys quantum error control, focusing on quantum stabilizer codes, stressing on the how to use classical codes to design good quantum codes.  ...  The set of error operators E n := {i λ w 1 ⊗ w 2 ⊗ . . . ⊗ w n } is a non-abelian group of cardinality 4 n+1 .  ... 
arXiv:2009.05735v1 fatcat:75uw6vlsdbgjjlpkejhqgcse6q

Coding Theory

Joachim Rosenthal, Mohammad Amin Shokrollahi
2007 Oberwolfach Reports  
The workshop on coding theory covered many facets of the recent research advances.  ...  Coding theory lies naturally at the intersection of a large number of disciplines in pure and applied mathematics: algebra and number theory, probability theory and statistics, communication theory, discrete  ...  Self-dual codes and invariant theory Gabriele Nebe (joint work with Eric Rains and Neil Sloane) In our joint book [2] we formalize the notion of a Type of a self-dual code.  ... 
doi:10.4171/owr/2007/56 fatcat:ac2rttsrn5g2rn5la72nudiyum

From Doubled Chern-Simons-Maxwell Lattice Gauge Theory to Extensions of the Toric Code [article]

T. Z. Olesen, N. D. Vlasii, U.-J. Wiese
2015 arXiv   pre-print
The Hilbert space of the theory is a product of local Hilbert spaces, each associated with a link and the corresponding dual link.  ...  In the pure Chern-Simons limit of a large "photon" mass, this results in a Z(k)-symmetric variant of Kitaev's toric code, self-adjointly extended by the two non-dynamical background lattice gauge fields  ...  In particular, one may ask whether they also reduce to variants of the toric code with a discrete non-Abelian gauge group.  ... 
arXiv:1503.07023v1 fatcat:t6cgmy7ak5gdfcbcnpjbgmgtve

Spectra of Gauge Code Hamiltonians [article]

Simon Burton
2018 arXiv   pre-print
Using Perron-Frobenius theory we constrain the location of first and second eigenvalues among the blocks of the Hamiltonian.  ...  This allows us to numerically find the gap of some large instances of the 3D gauge color code, which is compared to other frustrated spin Hamiltonians.  ...  The author would like to thank Andrew Doherty, Steven Flammia, Michael Kastoryano, Jiannis Pachos and Dan Browne.  ... 
arXiv:1801.03243v1 fatcat:dalo5hpnjve4fo7f75bmujgtbq

Stabilizer codes for Open Quantum Systems [article]

Francisco Revson F. Pereira and Stefano Mancini and Giuliano G. La Guardia
2022 arXiv   pre-print
This is done by pursuing an extension of the stabilizer formalism beyond the celebrated group structure of Pauli error operators.  ...  Lindblad master equation describes the evolution of a large variety of open quantum systems. An important property of some open quantum systems is the existence of decoherence-free subspaces.  ...  Acknowledgments The authors acknowledge the funding from the European Union's Horizon 2020 research and innovation programme, under grant agreement QUARTET No 862644.  ... 
arXiv:2107.11914v3 fatcat:7whmtzc6kjh4ffvbx4v5inlgbq

From doubled Chern–Simons–Maxwell lattice gauge theory to extensions of the toric code

T.Z. Olesen, N.D. Vlasii, U.-J. Wiese
2015 Annals of Physics  
The Hilbert space of the theory is a product of local Hilbert spaces, each associated with a link and the corresponding dual link.  ...  In the pure Chern-Simons limit of a large "photon" mass, this results in a Z(k)-symmetric variant of Kitaev's toric code, self-adjointly extended by the two non-dynamical background lattice gauge fields  ...  In particular, one may ask whether they also reduce to variants of the toric code with a discrete non-Abelian gauge group.  ... 
doi:10.1016/j.aop.2015.06.020 fatcat:o4egqopupbg6lbbicrwlrcevza

Algebraic characterization of MDS group codes over cyclic groups

A.A. Zain, B.S. Rajan
1995 IEEE Transactions on Information Theory  
Finally, it is shown that the dual code of an MDS group code over CM, a cyclic group with M elements, is also an MDS group code.  ...  A matrix is associated with the k(n -k) defining endomorphisms of the code and necessary and sufficient conditions for this matrix to define an MDS code over cyclic groups is proved.  ...  Biglieri for sending the preprints of [5] and the related papers and gratefully acknowledge one of the anonymous reviewers for his critical comments which helped to improve this correspondence and some  ... 
doi:10.1109/18.476335 fatcat:32y3bkntzzhhxgw6z5dcyja3uu

Ungauging quantum error-correcting codes [article]

Aleksander Kubica, Beni Yoshida
2018 arXiv   pre-print
We develop the procedures of gauging and ungauging, reveal their operational meaning and propose their generalization in a systematic manner within the framework of quantum error-correcting codes.  ...  We apply our formalism to the three-dimensional gauge color code (GCC) and show that its codeword space is equivalent to the Hilbert space of six copies of Z_2 lattice gauge theory with 1-form symmetries  ...  In particular, we are grateful to Héctor Bombín for emphasizing a connection between the gauge color code and Z 2 lattice gauge theory, and to Sam Roberts and Stephen Bartlett for sharing the draft of  ... 
arXiv:1805.01836v1 fatcat:6g5plsgnrfd6hhszvimkl4sq64

PhD thesis: Homological Quantum Codes Beyond the Toric Code [article]

Nikolas P. Breuckmann
2018 arXiv   pre-print
We perform numerical simulations to analyze the performance of these codes using decoders based on local updates.  ...  Furthermore, we give evidence that these codes can be used for more overhead efficient storage as compared to the surface code by orders of magnitude.  ...  eigenspace of a subgroup S of the Pauli group which is abelian and which does not contain −I.  ... 
arXiv:1802.01520v1 fatcat:rrbxw235undqzn22p43baytnte

Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes [article]

David Aasen, Zhenghan Wang, Matthew B. Hastings
2022 arXiv   pre-print
We argue that a general framework to understand this code is to consider continuous adiabatic paths of gapped Hamiltonians and we give a conjectured description of the fundamental group and second and  ...  Then, realizing this in the case of the toric code, we turn this path back into a sequence of checks, constructing an automorphism code closely related to the honeycomb code.  ...  ACKNOWLEDGMENTS ZW is partially supported by NSF grants FRG DMS-1664351, CCF 2006463, and ARO MURI contract W911NF-20-1-0082. DA would like to thank R. Thorngren for enlightening conversations.  ... 
arXiv:2203.11137v2 fatcat:xks4kurbbfenjlgowflnbqa7ua
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