Constructions of covering sequences and 2D-sequences
Yeow Meng Chee, Tuvi Etzion, Hoang Ta, Van Khu Vu · Designs Codes and Cryptography · 2025
Abstract An ( n , R )-covering sequence is a cyclic sequence whose consecutive n -tuples form a code of length n and covering radius R . Using several construction methods improvements of the upper bounds on the length of such sequences for $$n \le 20$$ n ≤ 20 and $$1 \le R \le 3$$ 1 ≤ R ≤ 3 , are obtained. The definition is generalized in two directions. An ( n , m , R )-covering sequence code is a set of cyclic sequences of length m whose consecutive n -tuples form a code of length n and covering radius R . The definition is also generalized to arrays in which the $$m \times n$$ m × n sub-matrices form a covering code with covering radius R . We prove that asymptotically there are covering sequences that attain the sphere-covering bound up to a constant factor.