Bounds on Shortest de Bruijn Covering Codes: M(n,R,2)

Information on Covering Codes: here.
Information on de Bruijn Cycles: here.

Definition: A covering code of radius R is a set of binary n-words so that every binary n-word can be reached from one of the codewords by changing at most R bits.  A binary de Bruijn covering code of radius R is a binary string so that the set of words appearing as n consecutive symbols (with wrap-around) is a covering code of radius R.

All unlabelled table entries are from the paper De Bruijn Cycles for Covering Codes, Fan Chung and Joshua N. Cooper, 2003.

Letters denote other sources noted below.

R/n

2

3

4

5

6

7

1

2

2

6

8

12

22

2

1

2

2

2

8

10

3

1

1

2

2

2

2

4

1

1

1

2

2

2

5

1

1

1

1

2

2

6

1

1

1

1

1

2

R/n

8

9

10

11

12

13

1

32

62a – 90b

105 – 210b

180 – 500b

342 – 1000b

598 – 2400b

2

14

20

38

38 – 90b

62 – 180b

97 – 400b

3

6

12

16

20

34 – 40

34 – 110b

4

2

2

4

8

16

24

5

2

2

2

2

8

8

6

2

2

2

2

2

2

7

2

2

2

2

2

2

8

1

2

2

2

2

2

9

1

1

2

2

2

2

10

1

1

1

2

2

2

11

1

1

1

1

2

2

1. P. R. J. Östergård and U. Blass, On the size of optimal binary codes of length 9 and covering radius 1, IEEE Trans. Inform. Theory, 47 (2001), 2556-2557.

2. M. Cooke, personal communication.