Syndrome realizations of linear codes and systems

Behavioral realizations of linear codes or systems usually involve observed variables, hidden variables (states), and constraints. We study realizations that also involve syndrome variables, which are hidden variables that must equal zero in any valid trajectory. Syndrome realizations arise naturally in parity-check or transform realizations of codes, and more generally as duals of state realizations. The dual of a syndrome realization has a nice form in which state and syndrome variables trade places. On the other hand, it is shown that syndrome realizations are essentially the same as the restricted state realizations of Forney, G.D., Jr. (see IEEE Trans. Inf. Theory, vol.47, p.520-48, 2001), which, in some respects, appear to be more fundamental.