6 - The Hexacode

Summary

The hexacode is a 3-dimensional, length 6 code over GF₄, the Galois field of order 4, whose codewords are readily remembered. Each of these codewords represents 2⁶ codewords of C and thus we obtain the 4³.2⁶ = 2¹² codewords of C. Each element of GF₄ = {0, 1, ω, ω̄} is given an odd and an even interpretation as a 4-dimensional column vector (corresponding to the columns of the MOG) or its complement: Thus a hexacodeword [1, 0, 0, 1, ω, ω̄] would have a 4-vector corresponding to 1 in the first column of the MOG, 0 in the second, 0 in the third and so on. All entries must be even or all entries odd and the first five columns may be complemented arbitrarily; the sixth column must then be complemented or not so that in the even interpretation the number of entries in the top row is even, and in the odd interpretation it is odd. Once the reader has memorized the hexacodewords, he or she can work with the MOG without the need of a physical copy of Figure 3.2.