  In this work, cyclic-skew-cyclic codes and sum-rank BCH codes are introduced.
Cyclic-skew-cyclic codes are characterized as left ideals of a suitable
non-commutative finite ring, constructed using skew polynomials on top of
polynomials (or vice versa). Single generators of such left ideals are found,
and they are used to construct generator matrices of the corresponding codes.
The notion of defining set is introduced, using pairs of roots of skew
polynomials on top of poynomials. A lower bound (called sum-rank BCH bound) on
the minimum sum-rank distance is given for cyclic-skew-cyclic codes whose
defining set contains certain consecutive pairs. Sum-rank BCH codes, with
prescribed minimum sum-rank distance, are then defined as the largest
cyclic-skew-cyclic codes whose defining set contains such consecutive pairs.
The defining set of a sum-rank BCH code is described, and a lower bound on its
dimension is obtained. Thanks to it, tables are provided showing that sum-rank
BCH codes beat previously known codes for the sum-rank metric for binary $ 2
\times 2 $ matrices (i.e., codes whose codewords are lists of $ 2 \times 2 $
binary matrices, for a wide range of list lengths that correspond to the code
length). Finally, a decoder for sum-rank BCH codes up to half their prescribed
distance is obtained.
