All Two-Weight Irreducible Cyclic Codes?

Authors:
Bernhard Schmidt;Clinton White
Affiliations:
Mathematisches Institut, Universität Augsburg, Universitätsstasse 15, 86135, Augsburg, Germanyf1schmidt@math.uni-augsburg.def1;Department of Mathematics, California Institute of Technology, Pasadena, California, 91125, f2cwhite@its.caltech.eduf2
Venue:
Finite Fields and Their Applications
Year:
2002

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Abstract

The aim of this paper is the classification of two-weight irreducible cyclic codes. Using Fourier transforms and Gauss sums, we obtain necessary and sufficient numerical conditions for an irreducible cyclic code to have at most two weights. This gives a unified explanation for all two-weight irreducible cyclic codes and allows a conjecturally complete classification. Aside from the two known infinite families of two-weight irreducible cyclic codes, a computer search reveals 11 sporadic examples. We conjecture that these are already all two-weight irreducible cyclic codes and give a partial proof of our conjecture conditionally on GRH.