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Level: BSc, MSci, MSc
Title: Lexicographic codes
Supervisor:

Dr Alex Fink
Research Area: Algebra
Description:

Several of our favourite small codes and families of codes over the field of order 2 can be constructed by a simple algorithm: run through a list of all binary strings of suitable length in lexicographic ("alphabetical") order, and add each in turn to the code if its distance from each previously selected codeword is long enough.  The project is to understand the construction, prove that in fact the set obtained this way is a code, and discuss some of the examples with reference to their appearance and special properties in other parts of mathematics, such as the Golay code, and/or connections to combinatorial games.

At MSc level, investigate how lexicodes behave as the string grows to infinity in both directions.

Further Reading:
  • J. H. Conway, N. J. A. Sloane, Lexicographic codes: error-correcting codes from game theory, IEEE Transactions on
    Information Theory     32 (1986) 337–348.
Key Modules:

MTH6108 Coding Theory

MTH6104 Algebraic Structures II (desirable)
Other Information:
Current Availability: Yes