Conference code[1][2; pg. 55]

Description

A member of the family of \((n,2n+2,(n-1)/2)\) nonlinear binary codes for \(n=1\) modulo 4 that are constructed from conference matrices.

A conference matrix \(H\) is a symmetric \(n+1\)-dimensional matrix with zero on its diagonal and \(\pm 1\) elsewhere that satisfies \(H H^T = n I_{n+1} \), where \(I_n\) is the \(n\)-dimensional identity matrix. By multiplying rows and columns by \(-1\), \(H\) can be normalized to the form \(\left(\begin{smallmatrix}0 & f\\ f^{T} & J \end{smallmatrix}\right)\), where \(J\) is an \(n\)-dimensional matrix satisfying \(J^{T}J=nI_{n}-F\) for some matrix \(F\) satisfying \(JF=FJ=0\). The code is made up of the \(2n\) rows of the two matrices \(\frac{1}{2}\left(I+F\pm J\right)\) along with the all-zeroes and all-ones vectors.