On the steiner triple systems of small rank embedded into perfect binary codes
D. I. Kovalevskaya, F. I. Solov’eva, E. S. Filimonova · Journal of Applied and Industrial Mathematics · 2013
Using the switching method, some classification is pointed out of STS(n) of order n = 2 r −1, r > 3, and small rank r n (different by 2 from the rank of the Hamming code of length n), embedded into perfect binary codes of length n and the same rank. The lower and upper bounds on the number of such different STSs are provided.Description of the class of STS(n)s of rank r n , which are not embedded into the perfect binary codes of length n and the same rank, and the lower bound on the number of these different systems are given. It is proved that each STS(n) of rank r n − 1 is embedded into a Vasil’ev code of length n and the same rank.