Steiner Systems From Binary Codes

The binary linear code of a Steiner triple system on \(2^d – 1\) points, where \(d \geq 3\) is an integer, contains a copy of the Hamming code \(\mathcal{H}_{di}\) this fact can be used to characterize those systems on \(2^d – 1\) points that have low dimension, and to show that these systems can always be extended to Steiner quadruple systems whose binary code is the extended code of the Steiner triple system.