Search

In this paper, we study problems about the similar configurations in F q d. Let G = (V , E) be a graph, where V = { 1 , 2 , ... , n } and E ⊆ ( V 2 ). For a set E in F q d , we say that E contains a pair of G with dilation ratio r if there exist distinct x 1 , x 2 , ... , x n ∈ E and distinct y 1 , y 2 , ... , y n ∈ E such that ‖ y i − y j ‖ = r ‖ x i − x j ‖ ≠ 0 whenever { i , j } ∈ E , where ‖ x ‖ : = x 1 2 + x 2 2 + ⋯ + x d 2 for x = (x 1 , x 2 , ... , x d) ∈ F q d. We show that if E has size at least C k q d / 2 , then E contains a pair of k -stars with dilation ratio r , and that if E has size at least C ⋅ min ⁡ { q (2 d + 1) / 3 , max ⁡ { q 3 , q d / 2 } } , then E contains a pair of 4-paths with dilation ratio r. Our method is based on enumerative combinatorics and graph theory. [ABSTRACT FROM AUTHOR]

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, two conjectures on permutation polynomials proposed recently by Wu and Li [19] are settled. Moreover, a new class of permutation trinomials of the form x + γ Tr q n / q ( x k ) is also presented, which generalizes two examples of [10] . [ABSTRACT FROM AUTHOR]

We study some sum-product problems over matrix rings. Firstly, for A , B , C ⊆ M n (F q) , we have | A + B C | ≳ q n 2 , whenever | A | | B | | C | ≳ q 3 n 2 − n + 1 2 . Secondly, if a set A in M n (F q) satisfies | A | ≥ C (n) q n 2 − 1 for some sufficiently large C (n) , then we have max ⁡ { | A + A | , | A A | } ≳ min ⁡ { | A | 2 q n 2 − n + 1 4 , q n 2 / 3 | A | 2 / 3 }. These improve the results due to The and Vinh (2020), and generalize the results due to Mohammadi, Pham, and Wang (2021). We also give a new proof for a recent result due to The and Vinh (2020). Our method is based on spectral graph theory and linear algebra. [ABSTRACT FROM AUTHOR]

Recently, Yaakobi et al. introduced codes for b -symbol read channels, where the read operation is performed as a consecutive sequence of b > 2 symbols. In this paper, we establish a Singleton-type bound for b -symbol codes. Codes meeting the Singleton-type bound are called maximum distance separable (MDS) codes, and they are optimal in the sense they attain the maximal minimum b -distance. We introduce a construction method using projective geometry, and then construct several infinite families of linear MDS b -symbol codes over finite fields. The lengths of these codes have a large range. And in some sense, we completely determine the existence of linear MDS b -symbol codes over finite fields for certain parameters. [ABSTRACT FROM AUTHOR]

In this paper, a new class of circulant matrices built from the deterministic filter and the deterministic subsampling is introduced for convolution-based compressed sensing. The pseudo-random sequences are applied in the construction of compressed sensing matrices. By using Katz's and Bombieri's character sum estimation, we are able to design good deterministic compressed sensing matrices for sparse recovery. In the worst case, the sparsity bound in our construction is similar to that of binary compressed sensing matrices constructed by DeVore and partial Fourier matrices constructed by Xu and Xu. Moreover, in the average case, we show that our construction can reconstruct almost all k -sparse vectors with m ≥ O ( k log ⁡ N ) . [ABSTRACT FROM AUTHOR]