The aim of this paper is twofold. First, we show a connection between the Lagrangian- Grassmannian variety geometry defined over a finite field with q elements and the q-ary Low-Density Parity- Check codes. Second, considering the Lagrangian-Grassmannian variety as a linear section of the Grassmannian variety, we prove that there is a unique linear homogeneous polynomials family, up to linear combination, such that annuls the set of its rational points.

In this work, we present some arithmetic properties of families of abelian [Formula: see text]-extensions of global function fields, among which are their generators and their type of ramification and decomposition.

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We apply a result of E. Kani relating genera and Hasse-Witt invariants of Galois extensions to a family of abelian p-extensions. Our formulas generalize the case of elementary abelian p-extensions found by Garcia and Stichtenoth.