Golay and other box codes

Abstract

The (24,12;8) extended Golay Code can be generated as a 6 x 4 binary matrix from the (15,11;3) BCH-Hamming Code, represented as a 5 x 3 matrix, by adding a row and a column, both of odd or even parity. The odd-parity case provides the additional 12th dimension. Furthermore, any three columns and five rows of the 6 x 4 Golay form a BCH-Hamming (15,11;3) Code. Similarly a (80,58;8) code can be generated as a 10 x 8 binary matrix from the (63,57;3) BCH-Hamming Code represented as a 9 x 7 matrix by adding a row and a column both of odd and even parity. Furthermore, any seven columns along with the top nine rows is a BCH-Hamming (53,57;3) Code. A (80,40;16) 10 x 8 matrix binary code with weight structure identical to the extended (80,40;16) Quadratic Residue Code is generated from a (63,39;7) binary cyclic code represented as a 9 x 7 matrix, by adding a row and a column, both of odd or even parity.