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Geometry, codes and difference sets: Exceptional connections. (English) Zbl 1032.94013
Arasu, K. T. (ed.) et al., Codes and designs. Proceedings of a conference honoring Professor Dijen K. Ray-Chaudhuri on the occasion of his 65th birthday, The Ohio State University, Columbus, OH, USA, May 18-21, 2000. Berlin: de Gruyter. Ohio State Univ. Math. Res. Inst. Publ. 10, 73-85 (2002).
The paper surveys some recent results on 2-error correcting BCH codes that tie together a number of areas, including cyclic codes, finite geometry and difference sets. An odd positive number \(s\) is called exceptional if the binary cyclic code \(C^{(m)}_s\) of length \(2^m- 1\) with zeros \(w\) and \(w^s\), where \(w\) is a primitive element of \(\text{GF}(2^m)\), is double-error-correcting for infinitely many \(m\). It is shown that \(C^{(m)}_s\) is double-error-correcting precisely when the reverse Dickson-Fibonacci polynomial \(h_s(z)= D_s(1, z)\) induces a one-to-one map on the set of elements of absolute trace \(0\) in \(\text{GF}(2^m)\). This characterization explains the known classes of exceptional \(s\) and provides a tool to bring to bear on the open conjectures of Janwa, McGuire and Wilson.
For the entire collection see [Zbl 0996.00030].

MSC:
94B15 Cyclic codes
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
94A55 Shift register sequences and sequences over finite alphabets in information and communication theory
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