Covering Sequences and Covering-Sequences Codes

An $(n,R)$-covering sequence is a cyclic sequence whose consecutive $n$-tuples form a code of length $n$ and covering radius $R$. An $(n,m,R)$-covering-sequences code is a set of cyclic sequences of length $m$, whose consecutive $n$-tuples form a code of length $n$ and covering radius $R$. These codes are the best building blocks for $(n,R)$-covering sequences. We show, for small radii, how the Hamming code can be used to construct such sequences of short length and such codes with a relatively small number of sequences and a total number of codewords. Sequences with small radius whose length approaches asymptotically to optimality are constructed, especially for an alphabet of prime power size large enough. With the same construction, interesting codes are also constructed for larger radii.