  In hypergraph theory, determining a characterization of the degree sequence
$d=(d_1,d_2,\ldots,d_n)$ where $d_1\ge d_2\ge\ldots,d_n$ are positive integers,
of an $h$-uniform simple hypergraph $\cal H$, and deciding the complexity
status of the reconstruction of $\cal H$ from $d$, are two challenging open
problems. They can be formulated in the context of discrete tomography: asks
whether there is a matrix $A$ with positive projection vectors
$H=(h,h,\ldots,h)$ and $V=(d_1,d_2,\ldots,d_n)$ with distinct rows. In this
paper we consider the two subcases where the vector $V$ is an homogeneous
vector, and where $V$ is almost homogeneous, i.e., $d_1-d_n=1$. We give a
simple characterization for these two subcases, and we show how to solve the
related reconstruction problems in polynomial time. To reach our goal, we use
the concepts of Lyndon words and necklaces of fixed density, and we apply some
already known algorithms for their efficient generation.
