Investigating the discrepancy property of de Bruijn sequences

Revisiting the Prefer-same and Prefer-opposite de Bruijn sequence constructions

“…Using a variant of the above discrepancy formula, Gabric and Sawada [10] numerically studied the discrepancy of a number of binary de Bruijn sequences, including the prefersame and prefer-opposite sequences which come out to have a low discrepancy. In this section we define a measure of discrepancy for sequences of general alphabet size and we use it to compare the prefer-higher sequence and the two proposed sequences with the binary versions.…”

“…Gabric and Sawada [10] conjecture that the prefer-opposite and prefer-same sequences have a discrepancy in the order of Θ(n 2 ). Table 3 below displays the calculated discrepancy for all three sequences for q = 2, .…”

Efficient constructions of the Prefer-same and Prefer-opposite de Bruijn sequences

“…Thus, CCR(w The CCR has been applied to efficiently construct de Bruijn sequences in variety of ways [11,24,20]. An especially efficient construction applies a concatenation scheme to construct a de Bruijn sequence with discrepancy (maximum difference between the number of 0s and 1s in any prefix) bounded above by 2n [18,19].…”

“…This makes the de Bruijn sequences S n and O n of special interest since they are, respectively, the lexicographically largest and smallest sequences with respect to a run-length encoding [3]. Moreover, recently it was noted they have a relatively small discrepancy when compared to the sequences generated by the Prefer-1 construction [19].…”