More box codes

Abstract

A new investigation shows that, starting from the BCH (21,15;3) code represented as a 7 x 3 matrix and adding a row and column to add even parity, one obtains an 8 x 4 matrix (32,15;8) code. An additional dimension is obtained by specifying odd parity on the rows and even parity on the columns, i.e., adjoining to the 8 x 4 matrix, the matrix, which is zero except for the fourth column (of all ones). Furthermore, any seven rows and three columns will form the BCH (21,15;3) code. This box code has the same weight structure as the quadratic residue and BCH codes of the same dimensions. Whether there exists an algebraic isomorphism to either code is as yet unknown.