By Peter Müller
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Additional info for Coding Theory, Edition: version 20 Jan 2014
The essential step in showing this is to prove that C is linear. See Exercise ??? for this. 2 The Ternary Golay Code In this section we construct the two ternary Golay codes. 8. Set 1 1 1 1 1 0 0 1 −1 −1 1 −1 1 0 1 − 1 − 1 −1 5×5 ∈ F6 × 6 . A = −1 1 0 1 −1 ∈ F3 and C = 3 A − 1 −1 −1 1 0 1 −1 1 −1 −1 1 0 −1 Then CC t = − I6 . Proof. Let ai be the i-th row of A. Then ai | a j = 1 if i = j, and ai | a j = −1 for i = j. As each row in A arises from rotating the preceding row to the right by one step, we may assume i = 1 in verifying the cases.
See [CHLL97] for a whole book devoted to covering codes. It also contains more details about Virtakallio’s exciting discovery. 3 Geometric Goppa Codes to be written 9 Appendix: Some Tools from Algebra to be written 36 Index binary Golay code, 29, 32 character, 5 conference matrix, 33 covering code, 34 covering radius, 12 distance distribution, 9 distance enumerator, 9 doubly-even, 20 dual code, 4 dual linear program, 11 extended binary Golay code, 29 extended ternary Golay code, 33 finite projective plane, 25 infeasible, 11 Krawtchouk polynomial, 14 Leech lattice, 32 linear character, 5 lines, 25 MacWilliams transformation, 12 order, 25 Paley matrix, 33 points, 25 projective plane, 25 self-dual, 20 Steiner systems, 31 ternary Golay code, 34 unbounded, 11 weight distribution, 8 weight enumerator, 8 37 References [Bes83] M.
Thus | G | = 24 · 23 · 22 · 21 · 20 · 48 = 244823040. Furthermore, one can show that G is simple, it is called the Mathieu group M24 , one of the 26 sporadic simple groups. The code G24 is connected to other fascinating combinatorial objects, the Steiner systems. Let 2 ≤ t < k < n be integers. A Steiner system S(t, k, n) is a set S of k–element subsets of a set M of size n, such that each t–element subset of M is a subset of exactly one set from S. For instance, a projective plane of order n gives an S(2, n + 1, n2 + n + 1) Steiner system, because any two distinct points are contained in exactly one line (which has size n + 1).