On interplays between Error-Correcting Codes and t-Designs

PhD Qualifying Examination


Title: "On interplays between Error-Correcting Codes and t-Designs"

by

Mr. Hao LIU


Abstract:

The theory of error-correcting code origins from the encoding problem in 
digital communication around 1950. Theoretical research on interplays between 
algebra error-correcting code and combinatoric t-design began to thrive from 
1970s. The first breakthrough in generating designs from linear codes was the 
Assmus-Mattson theorem. After revealing Z4 forms of the Kerdock and Preparata 
codes , research was also loaded on designs from Z4 codes. An analogue of 
Assmus-Mattson theorem in the Z4 case was also developed in 2010.

Analysis on linear codes from incidence matrices of designs became abundant in 
1 990s. A variety of design families were studied for generating linear codes, 
including yet not limited to planes, finite geometry designs, Hadamard 
difference sets and Steiner systems. A book named "Designs and Their Codes" by 
E.F. Assmus and J.D. Key is a nice summary on relevant results. A related topic 
is generating codes from difference sets, a combinatoric structure whose 
development is a symmetric 2-design. C. Ding summarise known results on 
corresponding codes in a wide range.

This survey gives brief conclusive descriptions on the development and 
achievements of both areas above. We firstly provide fundamental materials on 
algebra error-correcting codes and t-designs. Then we have discussions on 
general and detailed results on generating codes from designs and the other way 
round separately. Lastly, we draw a conclusion and bring up several open 
questions for future research.

Kew Words : Error-correcting code; t-Design;


Date:			Wednesday, 2 December 2015

Time:                  	10:00am - 12:00noon

Venue:                  Room 5510
                         Lifts 25/26

Committee Members:	Prof. Cunsheng Ding (Supervisor)
 			Prof. Huamin Qu (Chairperson)
 			Dr. Ke Yi
 			Dr. Maosheng Xiong (MATH)


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