Group rings for communications

Ted Hurley

National University of Ireland Galway

Abstract algebra

Abstract algebraic structures, and in particular group rings,frequently occur within the communications’ areas.

Engineers/Computer Scientists design examples of such structureswithout realising the significance of the general abstract structurewithin which the examples reside.

Better designs may be obtained using abstract algebra structures,in particular group rings, and required systems which have resisteddesign by ad-hoc methods can be constructed by abstract algebramethods.

Abstract algebra

Abstract algebraic structures, and in particular group rings,frequently occur within the communications’ areas.

Engineers/Computer Scientists design examples of such structureswithout realising the significance of the general abstract structurewithin which the examples reside.

Better designs may be obtained using abstract algebra structures,in particular group rings, and required systems which have resisteddesign by ad-hoc methods can be constructed by abstract algebramethods.

Examples

For example a code or filterbank that behaves in a certain way maybe required; the mathematician supplies the algebra that he/sheknows, from theory, will produce the code or filterbank to therequired specifications.

A particular case is where codes for implanted medical devices wererequired. Such devices require low storage and low power and thusa code stored by an algebraic formula which could generate thecode was the solution.

The matrix size could be 500 × 1000 but an algebraic expression toproduce the code may only require storing only 4 or 5 elements;this produces a code not only stored by an algebraic formula butalso what is called a Low Density Parity Check (LDPC) code andthese types are ‘known’ to perform well in practice.

Examples

For example a code or filterbank that behaves in a certain way maybe required; the mathematician supplies the algebra that he/sheknows, from theory, will produce the code or filterbank to therequired specifications.

A particular case is where codes for implanted medical devices wererequired. Such devices require low storage and low power and thusa code stored by an algebraic formula which could generate thecode was the solution.

The matrix size could be 500 × 1000 but an algebraic expression toproduce the code may only require storing only 4 or 5 elements;this produces a code not only stored by an algebraic formula butalso what is called a Low Density Parity Check (LDPC) code andthese types are ‘known’ to perform well in practice.

Examples

For example a code or filterbank that behaves in a certain way maybe required; the mathematician supplies the algebra that he/sheknows, from theory, will produce the code or filterbank to therequired specifications.

A particular case is where codes for implanted medical devices wererequired. Such devices require low storage and low power and thusa code stored by an algebraic formula which could generate thecode was the solution.

The matrix size could be 500 × 1000 but an algebraic expression toproduce the code may only require storing only 4 or 5 elements;this produces a code not only stored by an algebraic formula butalso what is called a Low Density Parity Check (LDPC) code andthese types are ‘known’ to perform well in practice.

Areas of application

Communications’ areas where group ring structures are extremelyuseful, and indeed at times prove indispensable, include thefollowing:

I Coding Theory. Coding theory may be thought of as the safetransfer of data and includes error-correcting code design.

More specifically, group rings excel in the construction of

1. Low Density parity check (LDPC) codes;2. Convolutional codes;3. Self-dual and dual-containing codes;4. Maximum and near maximum distance separable codes.

I Cryptography. This may be thought of as the securetransmission of data.

I Coding and Cryptography together. (Involves safetransmission and secure transmission together.)

Areas of application

Communications’ areas where group ring structures are extremelyuseful, and indeed at times prove indispensable, include thefollowing:

I Coding Theory. Coding theory may be thought of as the safetransfer of data and includes error-correcting code design.More specifically, group rings excel in the construction of

1. Low Density parity check (LDPC) codes;2. Convolutional codes;3. Self-dual and dual-containing codes;4. Maximum and near maximum distance separable codes.

I Cryptography. This may be thought of as the securetransmission of data.

I Coding and Cryptography together. (Involves safetransmission and secure transmission together.)

Areas of application

Communications’ areas where group ring structures are extremelyuseful, and indeed at times prove indispensable, include thefollowing:

I Coding Theory. Coding theory may be thought of as the safetransfer of data and includes error-correcting code design.More specifically, group rings excel in the construction of

1. Low Density parity check (LDPC) codes;2. Convolutional codes;3. Self-dual and dual-containing codes;4. Maximum and near maximum distance separable codes.

I Cryptography. This may be thought of as the securetransmission of data.

I Coding and Cryptography together. (Involves safetransmission and secure transmission together.)

Areas of application, continued ..

I Signal processing (filterbanks, wavelets). (Think of this as theprocessing of signals so as to eliminate noise; but it’s morethan that.)

I Multiple antenna code design. Think of this as transmittingsignals between multiple antennas; the word mimo, multipleinput-multiple output, is used. This has importantapplications in for example mobile phone communications.

I Compressed sensing. (To be explained later.) This hasnumerous applications.

I Software engineering.

I Threshold functions.

I ....

Remarks on applications

As we (now!) know, group rings have numerous applications in thecommunications’ areas. Much more can still be developed.

Should we say ‘sorry about that’ or feel embarrassed about theapplications?!

Mathematicians who have an aversion to applications should alsobe happy! There are indeed some very many nice theoremsinvolved.

In addition, the study of such applications gives great insight intothe structures of group rings themselves.

Also:

The funding organisations love this type of activity!

Remarks on applications

As we (now!) know, group rings have numerous applications in thecommunications’ areas. Much more can still be developed.

Should we say ‘sorry about that’ or feel embarrassed about theapplications?!

Mathematicians who have an aversion to applications should alsobe happy! There are indeed some very many nice theoremsinvolved.

In addition, the study of such applications gives great insight intothe structures of group rings themselves.

Also:

The funding organisations love this type of activity!

Remarks on applications

As we (now!) know, group rings have numerous applications in thecommunications’ areas. Much more can still be developed.

Should we say ‘sorry about that’ or feel embarrassed about theapplications?!

Mathematicians who have an aversion to applications should alsobe happy! There are indeed some very many nice theoremsinvolved.

In addition, the study of such applications gives great insight intothe structures of group rings themselves.

Also:

The funding organisations love this type of activity!

Remarks on applications

As we (now!) know, group rings have numerous applications in thecommunications’ areas. Much more can still be developed.

Should we say ‘sorry about that’ or feel embarrassed about theapplications?!

Mathematicians who have an aversion to applications should alsobe happy! There are indeed some very many nice theoremsinvolved.

In addition, the study of such applications gives great insight intothe structures of group rings themselves.

Also:

The funding organisations love this type of activity!

Eng/CS/IT

Engineers/Computer Science people get very little abstractalgebra. This is probably true for many (all?) Engineering andComputer Science programmes.

It is also clear that Engineering and Computer Science courseshave now less Mathematics courses than ever and little if any moreadvanced Algebra courses.

Thus engineers/cs people are unable to get to the cutting edge inmany areas which now require abstract algebra. Many are verycapable people and well able to cope with the ideas, given the rightbackground.

Eng/CS/IT

Engineers/Computer Science people get very little abstractalgebra. This is probably true for many (all?) Engineering andComputer Science programmes.

It is also clear that Engineering and Computer Science courseshave now less Mathematics courses than ever and little if any moreadvanced Algebra courses.

Thus engineers/cs people are unable to get to the cutting edge inmany areas which now require abstract algebra. Many are verycapable people and well able to cope with the ideas, given the rightbackground.

Underdetermined systems; a linear algebra problem, groupring approach.

I’ll briefly explain one of the areas mentioned.

Consider a system of equations Aw = y where A is an m × nmatrix, w an n × 1 unknown vector and u entries of y are knownwith u < m.

However it is given that w has at most t non-zero entries and thatu ≥ 2t.

Thus the vector w = (α1, α2, . . . , αn)T is known to have at most tnon-zero entries but the positions and the values of these non-zeroentries are unknown.

The system then should have a unique solution; find it.

Underdetermined systems; a linear algebra problem, groupring approach.

I’ll briefly explain one of the areas mentioned.

Consider a system of equations Aw = y where A is an m × nmatrix, w an n × 1 unknown vector and u entries of y are knownwith u < m.

However it is given that w has at most t non-zero entries and thatu ≥ 2t.

Thus the vector w = (α1, α2, . . . , αn)T is known to have at most tnon-zero entries but the positions and the values of these non-zeroentries are unknown.

The system then should have a unique solution; find it.

Underdetermined systems; a linear algebra problem, groupring approach.

I’ll briefly explain one of the areas mentioned.

Consider a system of equations Aw = y where A is an m × nmatrix, w an n × 1 unknown vector and u entries of y are knownwith u < m.

However it is given that w has at most t non-zero entries and thatu ≥ 2t.

Thus the vector w = (α1, α2, . . . , αn)T is known to have at most tnon-zero entries but the positions and the values of these non-zeroentries are unknown.

The system then should have a unique solution; find it.

Compressed sensing

This is known as compressed sensing for which there is a huge,extensive and expanding literature.

‘Compressed sensing is a signal processing technique for efficientlyacquiring and reconstructing a signal, by finding solutions tounderdetermined linear systems.’

Applications include MRI scanning, camera imaging and manymore.

The work by Candes, Romberg and Tao, is a basic reference forrecent treatments of compressed sensing.

Read Terence Tao’s blog on compressed sensing for a really nicebackground on it all.

Compressed sensing

This is known as compressed sensing for which there is a huge,extensive and expanding literature.

‘Compressed sensing is a signal processing technique for efficientlyacquiring and reconstructing a signal, by finding solutions tounderdetermined linear systems.’

Applications include MRI scanning, camera imaging and manymore.

The work by Candes, Romberg and Tao, is a basic reference forrecent treatments of compressed sensing.

Read Terence Tao’s blog on compressed sensing for a really nicebackground on it all.

Compressed sensing

This is known as compressed sensing for which there is a huge,extensive and expanding literature.

‘Compressed sensing is a signal processing technique for efficientlyacquiring and reconstructing a signal, by finding solutions tounderdetermined linear systems.’

Applications include MRI scanning, camera imaging and manymore.

The work by Candes, Romberg and Tao, is a basic reference forrecent treatments of compressed sensing.

Read Terence Tao’s blog on compressed sensing for a really nicebackground on it all.

Compressed sensing

This is known as compressed sensing for which there is a huge,extensive and expanding literature.

‘Compressed sensing is a signal processing technique for efficientlyacquiring and reconstructing a signal, by finding solutions tounderdetermined linear systems.’

Applications include MRI scanning, camera imaging and manymore.

The work by Candes, Romberg and Tao, is a basic reference forrecent treatments of compressed sensing.

Read Terence Tao’s blog on compressed sensing for a really nicebackground on it all.

It’s an error vector!

One approach taken is a linear algebra approach based onerror-correcting codes from group rings.

In Aw = y think of w , which has at most t non-zero entries, asthe error vector of a code; as long as the code can correct t errors,what is required then is a method to locate and determine these‘errors’.

It’s an error vector!

One approach taken is a linear algebra approach based onerror-correcting codes from group rings.

In Aw = y think of w , which has at most t non-zero entries, asthe error vector of a code; as long as the code can correct t errors,what is required then is a method to locate and determine these‘errors’.

Nicely said

“A large part of mathematics which becomes useful developed withabsolutely no desire to be useful, and in a situation where nobodycould possibly know in what area it would become useful; andthere were no general indications that it ever would be so.”

He continued: “By and large it is uniformly true in mathematicsthat there is a time lapse between a mathematical discovery andthe moment when it is useful; and that this lapse of time can beanything from 30 to 100 years, in some cases even more; and thatthe whole system seems to function without any direction, withoutany reference to usefulness, and without any desire to do thingswhich are useful.”

Think of Field Theory, Boolean Algebra, Number Theory ...

Janos von Neumann.

This ‘lapse of time’ can be much shorter these days.

Nicely said

“A large part of mathematics which becomes useful developed withabsolutely no desire to be useful, and in a situation where nobodycould possibly know in what area it would become useful; andthere were no general indications that it ever would be so.”

He continued: “By and large it is uniformly true in mathematicsthat there is a time lapse between a mathematical discovery andthe moment when it is useful; and that this lapse of time can beanything from 30 to 100 years, in some cases even more; and thatthe whole system seems to function without any direction, withoutany reference to usefulness, and without any desire to do thingswhich are useful.”

Think of Field Theory, Boolean Algebra, Number Theory ...

Janos von Neumann.

This ‘lapse of time’ can be much shorter these days.

Nicely said

“A large part of mathematics which becomes useful developed withabsolutely no desire to be useful, and in a situation where nobodycould possibly know in what area it would become useful; andthere were no general indications that it ever would be so.”

He continued: “By and large it is uniformly true in mathematicsthat there is a time lapse between a mathematical discovery andthe moment when it is useful; and that this lapse of time can beanything from 30 to 100 years, in some cases even more; and thatthe whole system seems to function without any direction, withoutany reference to usefulness, and without any desire to do thingswhich are useful.”

Think of Field Theory, Boolean Algebra, Number Theory ...

Janos von Neumann.

This ‘lapse of time’ can be much shorter these days.

Nicely said

“A large part of mathematics which becomes useful developed withabsolutely no desire to be useful, and in a situation where nobodycould possibly know in what area it would become useful; andthere were no general indications that it ever would be so.”

He continued: “By and large it is uniformly true in mathematicsthat there is a time lapse between a mathematical discovery andthe moment when it is useful; and that this lapse of time can beanything from 30 to 100 years, in some cases even more; and thatthe whole system seems to function without any direction, withoutany reference to usefulness, and without any desire to do thingswhich are useful.”

Think of Field Theory, Boolean Algebra, Number Theory ...

Janos von Neumann.

This ‘lapse of time’ can be much shorter these days.

Any references?

I ‘Algebraic structures for communications’, Contemp.Math,AMS, 611, 59-79, 2014.

I ‘Group rings for communications’, Int. J. Group Theory, Vol4, No. 4, 1-13, 2015.

There are related research papers, which are obtainable (in someperhaps earlier form) on ArXiv.

For example on the compressed sensing topic, discussed briefly, wehave:

’Solving underdetermined systems with error-correcting codes’, J.Information and Coding Theory, Vol. 4, No. 4, pp.201-221, 2017.

Any references?

I ‘Algebraic structures for communications’, Contemp.Math,AMS, 611, 59-79, 2014.

I ‘Group rings for communications’, Int. J. Group Theory, Vol4, No. 4, 1-13, 2015.

There are related research papers, which are obtainable (in someperhaps earlier form) on ArXiv.

For example on the compressed sensing topic, discussed briefly, wehave:

’Solving underdetermined systems with error-correcting codes’, J.Information and Coding Theory, Vol. 4, No. 4, pp.201-221, 2017.

Any references?

I ‘Algebraic structures for communications’, Contemp.Math,AMS, 611, 59-79, 2014.

I ‘Group rings for communications’, Int. J. Group Theory, Vol4, No. 4, 1-13, 2015.

There are related research papers, which are obtainable (in someperhaps earlier form) on ArXiv.

For example on the compressed sensing topic, discussed briefly, wehave:

’Solving underdetermined systems with error-correcting codes’, J.Information and Coding Theory, Vol. 4, No. 4, pp.201-221, 2017.