  The girth of a matrix is the least number of linearly dependent columns, in
contrast to the rank which is the largest number of linearly independent
columns. This paper considers the construction of {\it high-girth} matrices,
whose probabilistic girth is close to its rank. Random matrices can be used to
show the existence of high-girth matrices with constant relative rank, but the
construction is non-explicit. This paper uses a polar-like construction to
obtain a deterministic and efficient construction of high-girth matrices for
arbitrary fields and relative ranks. Applications to coding and sparse recovery
are discussed.
