Group rings for communications Ted Hurley National University of - - PowerPoint PPT Presentation

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 structures without realising the significance of the general abstract structure within which the examples reside.

Abstract algebra

Abstract algebraic structures, and in particular group rings, frequently occur within the communications’ areas. Engineers/Computer Scientists design examples of such structures without realising the significance of the general abstract structure within which the examples reside. Better designs may be obtained using abstract algebra structures, in particular group rings, and required systems which have resisted design by ad-hoc methods can be constructed by abstract algebra methods.

Examples

For example a code or filterbank that behaves in a certain way may be required; the mathematician supplies the algebra that he/she knows, from theory, will produce the code or filterbank to the required specifications.

Examples

For example a code or filterbank that behaves in a certain way may be required; the mathematician supplies the algebra that he/she knows, from theory, will produce the code or filterbank to the required specifications. A particular case is where codes for implanted medical devices were

• required. Such devices require low storage and low power and thus

a code stored by an algebraic formula which could generate the code was the solution.

Examples

For example a code or filterbank that behaves in a certain way may be required; the mathematician supplies the algebra that he/she knows, from theory, will produce the code or filterbank to the required specifications. A particular case is where codes for implanted medical devices were

• required. Such devices require low storage and low power and thus

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

Areas of application

Communications’ areas where group ring structures are extremely useful, and indeed at times prove indispensable, include the following:

◮ Coding Theory. Coding theory may be thought of as the safe

transfer of data and includes error-correcting code design.

Areas of application

Communications’ areas where group ring structures are extremely useful, and indeed at times prove indispensable, include the following:

◮ Coding Theory. Coding theory may be thought of as the safe

transfer 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.

Areas of application

Communications’ areas where group ring structures are extremely useful, and indeed at times prove indispensable, include the following:

◮ Coding Theory. Coding theory may be thought of as the safe

transfer 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.

◮ Cryptography. This may be thought of as the secure

transmission of data.

◮ Coding and Cryptography together. (Involves safe

transmission and secure transmission together.)

Areas of application, continued ..

◮ Signal processing (filterbanks, wavelets). (Think of this as the

processing of signals so as to eliminate noise; but it’s more than that.)

◮ Multiple antenna code design. Think of this as transmitting

signals between multiple antennas; the word mimo, multiple input-multiple output, is used. This has important applications in for example mobile phone communications.

◮ Compressed sensing. (To be explained later.) This has

numerous applications.

◮ Software engineering. ◮ Threshold functions. ◮ ....

Remarks on applications

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

Remarks on applications

As we (now!) know, group rings have numerous applications in the communications’ areas. Much more can still be developed. Should we say ‘sorry about that’ or feel embarrassed about the applications?!

Remarks on applications

As we (now!) know, group rings have numerous applications in the communications’ areas. Much more can still be developed. Should we say ‘sorry about that’ or feel embarrassed about the applications?! Mathematicians who have an aversion to applications should also be happy! There are indeed some very many nice theorems involved. In addition, the study of such applications gives great insight into the structures of group rings themselves. Also:

Remarks on applications

As we (now!) know, group rings have numerous applications in the communications’ areas. Much more can still be developed. Should we say ‘sorry about that’ or feel embarrassed about the applications?! Mathematicians who have an aversion to applications should also be happy! There are indeed some very many nice theorems involved. In addition, the study of such applications gives great insight into the structures of group rings themselves. Also: The funding organisations love this type of activity!

Eng/CS/IT

Engineers/Computer Science people get very little abstract

• algebra. This is probably true for many (all?) Engineering and

Computer Science programmes. It is also clear that Engineering and Computer Science courses have now less Mathematics courses than ever and little if any more advanced Algebra courses.

Eng/CS/IT

Engineers/Computer Science people get very little abstract

• algebra. This is probably true for many (all?) Engineering and

Computer Science programmes. It is also clear that Engineering and Computer Science courses have now less Mathematics courses than ever and little if any more advanced Algebra courses. Thus engineers/cs people are unable to get to the cutting edge in many areas which now require abstract algebra. Many are very capable people and well able to cope with the ideas, given the right background.

Underdetermined systems; a linear algebra problem, group ring approach.

I’ll briefly explain one of the areas mentioned.

Underdetermined systems; a linear algebra problem, group ring approach.

I’ll briefly explain one of the areas mentioned. Consider a system of equations Aw = y where A is an m × n matrix, w an n × 1 unknown vector and u entries of y are known with u < m.

Underdetermined systems; a linear algebra problem, group ring approach.

I’ll briefly explain one of the areas mentioned. Consider a system of equations Aw = y where A is an m × n matrix, w an n × 1 unknown vector and u entries of y are known with u < m. However it is given that w has at most t non-zero entries and that u ≥ 2t. Thus the vector w = (α1, α2, . . . , αn)T is known to have at most t non-zero entries but the positions and the values of these non-zero entries 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

This is known as compressed sensing for which there is a huge, extensive and expanding literature. ‘Compressed sensing is a signal processing technique for efficiently acquiring and reconstructing a signal, by finding solutions to underdetermined linear systems.’

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 efficiently acquiring and reconstructing a signal, by finding solutions to underdetermined linear systems.’ Applications include MRI scanning, camera imaging and many more. The work by Cand` es, Romberg and Tao, is a basic reference for recent treatments of compressed sensing.

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 efficiently acquiring and reconstructing a signal, by finding solutions to underdetermined linear systems.’ Applications include MRI scanning, camera imaging and many more. The work by Cand` es, Romberg and Tao, is a basic reference for recent treatments of compressed sensing. Read Terence Tao’s blog on compressed sensing for a really nice background on it all.

It’s an error vector!

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

It’s an error vector!

One approach taken is a linear algebra approach based on error-correcting codes from group rings. In Aw = y think of w, which has at most t non-zero entries, as the 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 with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so.”

Nicely said

“A large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so.” He continued: “By and large it is uniformly true in mathematics that there is a time lapse between a mathematical discovery and the moment when it is useful; and that this lapse of time can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful.” Think of Field Theory, Boolean Algebra, Number Theory ...

Nicely said

“A large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so.” He continued: “By and large it is uniformly true in mathematics that there is a time lapse between a mathematical discovery and the moment when it is useful; and that this lapse of time can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful.” Think of Field Theory, Boolean Algebra, Number Theory ... J´ anos von Neumann.

Nicely said

“A large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so.” He continued: “By and large it is uniformly true in mathematics that there is a time lapse between a mathematical discovery and the moment when it is useful; and that this lapse of time can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful.” Think of Field Theory, Boolean Algebra, Number Theory ... J´ anos von Neumann. This ‘lapse of time’ can be much shorter these days.

Any references?

◮ ‘Algebraic structures for communications’, Contemp.Math,

AMS, 611, 59-79, 2014.

◮ ‘Group rings for communications’, Int. J. Group Theory, Vol

4, No. 4, 1-13, 2015.

Any references?

◮ ‘Algebraic structures for communications’, Contemp.Math,

AMS, 611, 59-79, 2014.

◮ ‘Group rings for communications’, Int. J. Group Theory, Vol

4, No. 4, 1-13, 2015. There are related research papers, which are obtainable (in some perhaps earlier form) on ArXiv.

Any references?

◮ ‘Algebraic structures for communications’, Contemp.Math,

AMS, 611, 59-79, 2014.

◮ ‘Group rings for communications’, Int. J. Group Theory, Vol

4, No. 4, 1-13, 2015. There are related research papers, which are obtainable (in some perhaps earlier form) on ArXiv. For example on the compressed sensing topic, discussed briefly, we have: ’Solving underdetermined systems with error-correcting codes’, J. Information and Coding Theory, Vol. 4, No. 4, pp.201-221, 2017.