  A new construction for moderate density parity-check (MDPC) codes using
finite geometry is proposed. We design a parity-check matrix for this family of
binary codes as the concatenation of two matrices: the incidence matrix between
points and lines of the Desarguesian projective plane and the incidence matrix
between points and ovals of a projective bundle. A projective bundle is a
special collection of ovals which pairwise meet in a unique point. We determine
minimum distance and dimension of these codes, showing that they have a natural
quasi-cyclic structure. In addition, we analyze the error-correction
performance within one round of a modification of Gallager's bit-flipping
decoding algorithm. In this setting, our codes have the best possible
error-correction performance for this range of parameters.
