  We consider discrete Dirac systems as an alternative (to the famous Szeg\H{o}
recurrencies and matrix orthogonal polynomials) approach to the study of the
corresponding block Toeplitz matrices. We prove an analog of the
Christoffel--Darboux formula and derive the asymptotic relations for the analog
of reproducing kernel (using Weyl--Titchmarsh functions of discrete Dirac
systems). We study also the case of rational Weyl--Titchmarsh functions (and
GBDT version of the B\"acklund-Darboux transformation of the trivial discrete
Dirac system). We show that block diagonal plus block semi-separable Toeplitz
matrices appear in this case.
