Narrow Results

We generalize a result of Flenner, proved in characteristic zero, to positive characteristics. We prove that the first syzygy bundle, , of the line bundle over is semistable, for a certain infinite set of integers d ≥ 0. Moreover, for arbitrary d, there is a "good enough estimate" on in terms of d and n; thus a strong restriction theorem of Langer, proved earlier for characteristic k > d, is valid in arbitrary characteristics.

In this work, we analyze the problem of catastrophicity of encoders of convolutional codes over the Laurent series with coefficients in ${\mathbb{Z}}_{p^r}$, ${\mathbb{Z}}_{p^r}((d))$. Kuijper and Pinto proved in [M. Kuijper and R. Pinto, On minimality of convolutional ring encoders, IEEE Trans. Autom. Control55(11) (2009) 4890–4897] that, contrary to what happens for codes over $𝔽((d))$, where $𝔽$ is a field, when dealing with ${\mathbb{Z}}_{p^r}((d))$ there are convolutional codes that do not admit non-catastrophic encoders. Nevertheless it was conjectured that any catastrophic convolutional code admits another type of non-catastrophic encoder called $p$-encoder. In this paper we solve this conjecture for a class of $(2,1)$ convolutional codes over ${\mathbb{Z}}_{p^2}$ and show that, in fact, these codes always admit a non-catastrophic p-encoder. We also describe a constructive procedure that allows us to obtain a non-catastrophic $p$-encoder.