Bounding the trellis state complexity of algebraic geometric codes
Let C be an algebraic geometric code of dimension k and length n constructed on a curve X over F-q. Let s (C) be the state complexity of C and set w(C) := min{k, n - k}, the Wolf upper bound on s(C). We introduce a numerical function R that depends on the gonality sequence of X and show that s(C) greater than or equal to w(C) - R(2g - 2), where g is the genus of X. As a matter of fact, R(2g - 2) less than or equal to g - (gamma(2) - 2) with gamma(2) being the gonality of X over F-q, and thus in particular we have that s (C) greater than or equal to (C) g + gamma(2) - 2.
trellis state complexity algebraic geometric codes error correcting codes gonality sequence of curves