New constructions for moderate-density parity-check (MDPC) codes using finite geometry are proposed. We design a parity-check matrix for the main 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 the minimum distance and the dimension of these codes, and we show that they have a natural quasi-cyclic structure. We consider alternative constructions based on an incidence matrix of a Desarguesian projective plane and compare their error-correction performance with regards to a modification of Gallager's bit-flipping decoding algorithm. In this setting, our codes have the best possible error-correction performance after one round of bit-flipping decoding given the parameters of the code's parity-check matrix.

We extend the original cylinder conjecture on point sets in affine three-dimensional space to the more general framework of divisible linear codes over${ {\mathbb {F}}_{q}}$and their classification. Through a mix of linear programming, combinatorial techniques and computer enumeration, we investigate the structural properties of these codes. In this way, we can prove a reduction theorem for a generalization of the cylinder conjecture, show some instances where it does not hold and prove its validity for small values of$q$. In particular, we correct a flawed proof for the original cylinder conjecture for$q = 5$and present the first proof for$q = 7$.

The notion of digits in finite fields was introduced a few years ago as an attempt to deliver insight in related yet unresolved questions over the natural numbers. Several such intractable questions are related to the sum of digits function, which assigns to every natural number the sum of its digits. Its analogue over finite fields, which is a map from $\mathbb{F}_q$ to $\mathbb{F}_p$, $q=p^h$ where $p$ prime and $h \geq 2$, has been studied by several authors. In particular, Cathy Swaenepoel [ J. Number Theory, 189 (2018), pp. 97--114] investigated the sum of digits of products of field elements. The main techniques involved were estimates on certain character sums and Gaussian sums over $\mathbb{F}_q$ and $\mathbb{F}_p$. In this paper, we extend and generalize these results using a different approach, based on spectral graph theory, without any reference to character theory.