Motivation: I am interested about the application of group theory to information theory. To be precise, I am interested in data compression (source coding theory).
Question: Is there any paper/survey paper on group-theoretic information theory?
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$\begingroup$ This is the closest thing I got physics.stackexchange.com/questions/30229/… $\endgroup$– MichaelCommented Jul 23, 2016 at 14:32
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$\begingroup$ Also, this book indicates but gives no reference books.google.com.bd/… $\endgroup$– MichaelCommented Jul 23, 2016 at 14:36
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2$\begingroup$ This recent STOC paper uses code symmetries to show RM codes achieve capacity in erasure channels: dl.acm.org/authorize?N04174. It doesn't use deep group theory, but it does use it in an essential way. $\endgroup$– Sasho NikolovCommented Jul 23, 2016 at 19:19
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$\begingroup$ @SashoNikolov there are many papers on coding theory and group theory but I think the scope of the post is within information theory $\endgroup$– TurboCommented Jul 23, 2016 at 23:42
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1$\begingroup$ @Jim if you are interested in coding theory start with Slepian's papers in the 50s and trace all the way to today $\endgroup$– TurboCommented Jul 23, 2016 at 23:44
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2 Answers
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Sadly, group structure is nearly so limited that there isn't much one can do with it to be of use in information theory, thus the literature is prone to be fairly sparse. Even Abelian groups aren't enough structure.
Even basic abstract algebra texts which have some basic coding theory applications generally provide examples using field theory or linear algebra. If there were useful examples for groups, they'd most likely be introduced there.
- Hungerford, Thomas; Abstract Algebra: An Introduction
- Rotman,J.J.; A First Course in Abstract Algebra, Third Edition
As most of Rotman's work was in group theory, if anyone would have had examples there, certainly he would. Steven Roman at UCI also did a good bit of textbook writing (separately) on coding theory and field theory and has a text Fundamentals of Group Theory: An Advanced Approach (Birkhauser), but not having read it, I suspect you'll find it barren of the group theoretical work you're looking for.
In the late 90's I seem to recall some work on binary codes defined by using codes over the alphabet $ \mathbb {Z}_4$. For some background (and possible references) see J.H. van Lint's Introduction to Coding Theory, Third Edition (Springer, 1999).
Once you take the algebraic step up to even finite fields, you're far more likely to find overlap with more coding theory. For this, try taking a look at Error-Correcting Linear Codes: Classification by Isometry and Applications (Springer) by Betten, Braun, et al. which may have some scant material on applications to groups as I recall.
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1$\begingroup$ how does this answer your post? it addresses channel coding not source coding. $\endgroup$– TurboCommented Aug 7, 2016 at 13:40
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Reference Goppa's information theory work.
[CITATION] Nonprobabilistic mutual information without memory VD Goppa - Probl. Contr. Inform. Theory, 1975
I know no other work which uses group theory to frame information theory.
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2$\begingroup$ Thanks, but Wiki says he used more advance algebraic structure than group, i.e. he uses field , I was looking for group only, to be precise symmetric group. I can't remember,probably S. Aaronson used Group theory in info: theory/network theory. Searching has not helped. $\endgroup$– MichaelCommented Jul 23, 2016 at 15:52