Constructions of covering sequences and 2D-sequences

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 $$n \le 20$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>20</mml:mn> </mml:mrow> </mml:math> and $$1 \le R \le 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>R</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , 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 $$m \times n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> 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.