<p>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.</p>

<p>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.</p>

<ul>
<li><a href="http://amzn.to/2aFmxPA" rel="noreferrer">Hungerford, Thomas; Abstract Algebra: An Introduction</a> </li>
<li><a href="http://amzn.to/2aBIVfe" rel="noreferrer">Rotman,J.J.; A First Course in Abstract Algebra, Third Edition</a></li>
</ul>

<p>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 <a href="http://amzn.to/2atcFoK" rel="noreferrer">Fundamentals of Group Theory: An Advanced Approach (Birkhauser)</a>, but not having read it, I suspect you'll find it barren of the group theoretical work you're looking for.</p>

<p>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 <a href="http://amzn.to/2aI4O7k" rel="noreferrer">J.H. van Lint's Introduction to Coding Theory, Third Edition (Springer, 1999)</a>.</p>

<p>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 <a href="http://amzn.to/2aAyrGJ" rel="noreferrer">Error-Correcting Linear Codes: Classification by Isometry and Applications (Springer) by Betten, Braun, et al.</a> which may have some scant material on applications to groups as I recall.</p>

CC BY-SA 3.0
2016-08-07T13:40:20.640Z
how does this answer your post? it addresses channel coding not source coding.
Turbo
CC BY-SA 3.0
2016-08-06T04:21:56.913Z
2016-08-06T04:21:56.913Z
Chris Aldrich
<p>Reference Goppa's information theory work.</p>

<p><a href="http://iopscience.iop.org/article/10.1070/RM1984v039n01ABEH003062/meta;jsessionid=2978C0F66C0E4C77833FEDFE7B511F98.c1.iopscience.cld.iop.org" rel="nofollow">http://iopscience.iop.org/article/10.1070/RM1984v039n01ABEH003062/meta;jsessionid=2978C0F66C0E4C77833FEDFE7B511F98.c1.iopscience.cld.iop.org</a></p>

<p>[CITATION] Nonprobabilistic mutual information without memory
VD Goppa - Probl. Contr. Inform. Theory, 1975</p>

<p>I know no other work which uses group theory to frame information theory.</p>

CC BY-SA 3.0
2016-07-23T15:52:01.333Z
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.
Michael
CC BY-SA 3.0
2016-07-23T15:04:56.267Z
2016-07-23T16:24:51.900Z
2016-07-23T16:24:51.900Z
Turbo
<p><strong>Motivation:</strong> I am interested about the application of group theory to information theory. To be precise, I am interested in data compression (source coding theory).</p>

<p><strong>Question:</strong>
Is there any paper/survey paper on group-theoretic information theory?</p>

CC BY-SA 3.0
2016-07-23T14:32:30.723Z
This is the closest thing I got http://physics.stackexchange.com/questions/30229/could-the-periodic-table-have-been-done-using-group-theory
Michael
CC BY-SA 3.0
2016-07-23T14:36:58.190Z
Also, this book indicates but gives no  reference https://books.google.com.bd/books?id=dtusau3dpkAC&pg=PA168&lpg=PA168&dq=data+compression+and+group+theory&source=bl&ots=L3EFOgMcI8&sig=81hwsV4OYssYOG75BisMvojEUBA&hl=en&sa=X&redir_esc=y#v=onepage&q=data%20compression%20and%20group%20theory&f=false
Michael
CC BY-SA 3.0
2016-07-23T19:19:00.990Z
This recent STOC paper uses code symmetries to show RM codes achieve capacity in erasure channels: http://dl.acm.org/authorize?N04174. It doesn't use *deep* group theory, but it does use it in an essential way.
Sasho Nikolov
CC BY-SA 3.0
2016-07-23T23:42:59.670Z
@SashoNikolov there are many papers on coding theory and group theory but I think the scope of the post is within information theory
Turbo
CC BY-SA 3.0
2016-07-23T23:44:05.240Z
@Jim if you are interested in coding theory start with Slepian's papers in the 50s and trace all the way to today
Turbo
CC BY-SA 3.0
2016-07-25T05:42:48.333Z
Specifically you could also look at tiling for studying shannon entropy, use penrose tilings (which are group theoretical objects) for a binary source  to fill a plane,  and can be used for studying the shannon entropy, sequences of 0s and 1s and then study their channel capacity.  the link to groups is weak as the symmetry in this case(penrose tilings) aperiodic. Groups imply an implicit symmetric structure. So aperiodic tilings for 00, ad 01 sequences, and base a channel capacity via markov chains could be one way to look for a link to group theory.
user3483902
CC BY-SA 3.0
2016-07-25T06:18:25.763Z
@user3483902 do you have references for the connections you are referring? It looks more like spectral theory and not group theory.
Turbo
<reference-request><it.information-theory><coding-theory><gr.group-theory>
Relation between group theory and information theory
Michael
# Relation between group theory and information theory

(Asked by: username_0 on 2016-07-23)

**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?

**Tags:** reference-request, it.information-theory, coding-theory, gr.group-theory

> **Comments:**
> **username_0**: This is the closest thing I got http://physics.stackexchange.com/questions/30229/could-the-periodic-table-have-been-done-using-group-theory  
> **username_0**: Also, this book indicates but gives no  reference https://books.google.com.bd/books?id=dtusau3dpkAC&pg=PA168&lpg=PA168&dq=data+compression+and+group+theory&source=bl&ots=L3EFOgMcI8&sig=81hwsV4OYssYOG75BisMvojEUBA&hl=en&sa=X&redir_esc=y#v=onepage&q=data%20compression%20and%20group%20theory&f=false  
> **username_1**: This recent STOC paper uses code symmetries to show RM codes achieve capacity in erasure channels: http://dl.acm.org/authorize?N04174. It doesn't use *deep* group theory, but it does use it in an essential way.  
> **username_2**: @SashoNikolov there are many papers on coding theory and group theory but I think the scope of the post is within information theory  
> **username_2**: @Jim if you are interested in coding theory start with Slepian's papers in the 50s and trace all the way to today  
> **username_3**: Specifically you could also look at tiling for studying shannon entropy, use penrose tilings (which are group theoretical objects) for a binary source  to fill a plane,  and can be used for studying the shannon entropy, sequences of 0s and 1s and then study their channel capacity.  the link to groups is weak as the symmetry in this case(penrose tilings) aperiodic. Groups imply an implicit symmetric structure. So aperiodic tilings for 00, ad 01 sequences, and base a channel capacity via markov chains could be one way to look for a link to group theory.  
> **username_2**: @username_3 do you have references for the connections you are referring? It looks more like spectral theory and not group theory.  

---

## 2 answers

### Answer by username_4 on 2016-08-06. Score: 7 | Accepted answer

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](http://amzn.to/2aFmxPA)
* [Rotman,J.J.; A First Course in Abstract Algebra, Third Edition](http://amzn.to/2aBIVfe)

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)](http://amzn.to/2atcFoK), 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)](http://amzn.to/2aI4O7k).

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.](http://amzn.to/2aAyrGJ) which may have some scant material on applications to groups as I recall.

> **Comments:**
> **username_2**: how does this answer your post? it addresses channel coding not source coding.  

---

### Answer by username_2 on 2016-07-23. Score: 4

Reference Goppa's information theory work.

<http://iopscience.iop.org/article/10.1070/RM1984v039n01ABEH003062/meta;jsessionid=2978C0F66C0E4C77833FEDFE7B511F98.c1.iopscience.cld.iop.org>

[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.

> **Comments:**
> **username_0**: 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.  

---

