Splitter Sets and $k$ -Radius Sequences

Short k‐radius sequences, k‐difference sequences and universal cycles

“…Lemma 2.5 (Bange et al [5], Bondy et al [8,9], Singer [37], and Zhang et al [42]). Zhang et al [42] also gave several explicit constructions of k-additive sequences by applying the Weil's theorem on multiplicative character sums. Lemma 2.6 (Zhang et al [42]).…”

“…Informally, any two distinct elements of Σ occur within distance k from each other somewhere in the sequence. For example, the following sequence from [42], 3, 6, 2, 7, 0, 5, 6, 4, 1, 0, 7, 3, 4, 2, 5, 1, 3 is an 8-ary 2-radius sequence of length 17 over {0, 1, 2, …, 7}.…”

“…The extreme problem of determining the shortest length is also interesting in pure combinatorics. In a recent paper [42], the authors revealed the connection of short k-radius sequences with the existence of splitter sets, which have been widely studied due to their applications in multilevel cell (MLC) flash memories [11,29,28,30,35,36,38,40,41]. Let f n ( ) k be the shortest length of an n-ary k-radius sequence.…”

“…For small integers n, or when n p = 2 and p > 2 is a prime, exact values of f n ( ) 2 can be found in [12,26,27]. For k = 3, 4, sparse values of f n ( ) k were determined in [42] by computer search. Recently, a novel construction was given in [8,9], where Bondy, Lonc, and Rzążewski connected the shortest k-radius sequences with newly defined sequences, which they call k-additive sequences.…”

“…In [8], Bondy, Lonc, and Rzążewski proposed the following conjecture, and confirmed it for k = 2. Very recently, the authors in [32,42] independently showed that: for every fixed k, there are infinitely many primes n such that f n On…”