 

Cyclotomics Logo Registered U.S. Trademark, April 10, 1984 This sketch of Fano's projective plane of order 2, equivalent to the Hamming (7,4) errorcorrecting code, was the logo of Cyclotomics, Inc., the company which Berlekamp founded and led in the early 1980s. It was acquired by Eastman Kodak in 1985. It continued another decade under the name of Kodak Berkeley Research. Mathematicians will be interested to note that the symmetry group of the celebrated Klein surface depicted in the sculpture in the Berlekamp garden at MSRI has symmetry group which is twice the symmetry group of the object depicted in Cyclotomics logo. The first published versions of the Hamming (7,4) code appeared in the papers of Shannon [1948] and Hamming [1949]. Mathematically, it is equivalent to a Venn diagram which had been published earlier by R. A. Fisher in his work on the design of experiments. The (7,4) Hamming code has 2^4 codewords, each 7 bits long. By annexing an overall parity check, one obtains the (8,4) binary Hamming code whose minimum distance is 4. The NordstromRobinson code, which has 2^8 binary codewords of length 16, may be viewed as an (8,4) code over Z4. Its Lee distance is 6, and it can readily be seen to contain multiple copies of the Hamming (8,4) code. The (24,12) Golay code can be viewed as the projection onto a particular basis of Z2 of a the (8,4) ReedSolomon code, and from that perspective it too can be seen to contain multiple copies of the Hamming (8,4) code. Finally, the Golay code was extended to the Leech lattice, whose symmetry group and its extensions to the "monster" group form the crucial subset of all sporadic finite simple groups.  