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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
##### Keywords:
BCH cyclic code; 2-error correcting codes