Title page for ETD etd-0713101-132954


Type of Document Dissertation
Author Guneri, Cem
Author's Email Address cguner1@lsu.edu
URN etd-0713101-132954
Title Artin-Schreier Families and 2-D Cyclic Codes
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Advisory Committee
Advisor Name Title
Robert Lax Committee Chair
Frank Neubrander Committee Member
Jurgen Hurrelbrink Committee Member
Neal Stoltzfus Committee Member
William Adkins Committee Member
Gil Lee Dean's Representative
Keywords
  • cyclic codes
  • artin-schreier
  • mathematics
Date of Defense 2001-07-26
Availability unrestricted
Abstract
We start with the study of certain Artin-Schreier families. Using coding theory techniques, we determine a necessary and sufficient condition for such families to have a nontrivial curve with the maximum possible number of rational points over the finite field in consideration. This result produces several nice corollaries, including the existence of certain maximal curves; i.e., curves meeting the Hasse-Weil bound.We then present a way to represent two-dimensional (2-D) cyclic codes as trace codes starting from a basic zero set of its dual code. This representation enables us to relate the weight of a codeword to the number of rational points on certain Artin-Schreier curves via the additive form of Hilbertís Theorem 90. We use our results on Artin-Schreier families to give a minimum distance bound for a large class of 2-D cyclic codes. Then, we look at some specific classes of 2-D cyclic codes that are not covered by our general result. In one case, we obtain the complete weight enumerator and show that these types of codes have two nonzero weights. In the other cases, we again give minimum distance bounds. We present examples, in some of which our estimates are fairly effcient.
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