  In this paper three Schur ring are discussed, namenly: Hamming, circulant
orbists and decimated circulant orbits Schur ring. By using autocorrelation
function and the run structure of binary sequences we proof the relation
between this Schur ring and combinatorial structures such as Hadamard matrices,
periodic compatible binary sequences and perfect binary sequences. Cai proved
for binary sequences that the autocorrelation function is in fact completely
determined by its run structure. Also, he characterised the structure of the
circulant Hadamard matrices. We characterise a more general structure, called
periodic compatible binary sequences ($PComS$ for brevety), which generalises
Hadamard matrices, periodic complementary binary sequences and binary sequences
with $2$-level autocorrelation. Families of periodic compatibles binary
sequences are presented. Also, we compute a bounds on familias $PComS$ in
Hamming Schur ring. The results obtained are applied to families of $PComS$
such as circulant, with one and two circulant cores, Goethals-Seidel type and
partial Hadamard matrices and perfect binary sequences.
