On the existence of perfect splitter sets

Abstract

Given integers k 1 , k 2 with 0 ≤ k 1 < k 2 , the determinations of all positive integers q for which there exists a perfect Splitter B[−k 1 , k 2 ](q) set is a wide open question in general. In this paper, we obtain new necessary and sufficient conditions for an odd prime p such that there exists a nonsingular perfect B[−1, 3](p) set. We also give some necessary conditions for the existence of purely singular perfect splitter sets. In particular, we determine all perfect B[−k 1 , k 2 ](2 n ) sets for any positive integers k 1 , k 2 with k 1 + k 2 ≥ 4. We also prove that there are infinitely many prime p such that there exists a perfect B[−1, 3](p) set.

Nonsingular splittings over finite fields

“…Lemma 4.2. ( [35], Theorem IV.1.) Let k 1 , k 2 be positive integers with 1 ≤ k 1 ≤ k 2 and let p be an odd prime with p ≡ 1(mod…”

“…In [35], we obtained a necessary and sufficient conditions for the prime p such that [−1, 3] * splits Z p with the some splitter set B. Now we give a presentation of B for [−1, 5] * , we have.…”

On lattice tilings of $\mathbb{Z}^{n}$ by limited magnitude error balls $\mathcal{B}(n,2,1,1)$

“…See [3], [6], [7], [8], [13], [14], [15], [17], [18], [19], [23] for the researches on cross B(n, 1, k, k) and semi-cross B(n, 1, k, 0). Later, these two shapes are extended to quasi-cross B(n, 1, k + , k − ) by Schwartz [11], and immediately, quasi-cross was received a lot of attentions [12], [24], [25], [27], [28], [29], [30], [31], [32]. For t ≥ 2, there are only a few results.…”

Some New Results on Splitter Sets

“…In [33]- [35], the authors proved that there does not exist a nonsingular perfect splitter set when 1 ≤ k 1 < k 2 and k 1 + k 2 is an odd integer. In [32], Yuan and Zhao gave a necessary and sufficient condition for the existence of nonsingular perfect B[−1, 3](p) sets. For k 1 = k 2 = 4, Tamm [27] provided a list of primes p for which a perfect B[−4, 4](p) set exists.…”