In multi-access coded caching (MACC) schemes, most research assumes that the users access z continuous caches by the way of a cyclic wraparound user-to-cache association. Recently, N. Das and B. S. Rajan identify a class of user-to-cache association bipartite graphs, which achieves either a lower transmission rate or subpacketization level than all other existing MACC schemes using a cyclic wrap-around user-to-cache association. In this paper, we define a class of user-to-cache association matrix, and using the matrix, we improve the MACC schemes proposed by N. Das and B. S. Rajan in [2]. The improved schemes can achieve a lower transmission rate than the original scheme of [2] with the same subpacketization level under some cases.

In quantum information theory, symmetric informationally complete positive operator-valued measures (SIC-POVMs) are related to quantum state tomography, quantum cryptography and foundational studies. It is difficult to construct SIC-POVMs and remains unknown whether SIC-POVMs exist for infinitely dimension. Therefore, researchers have proposed the solution to construct approximately symmetric informationally complete positive operator-valued measures (ASIC-POVMs). This paper proposes a construction of ASIC-POVMs using character sums over Galois rings. The dimension of this ASIC-POVMs is q -1, where q is a prime power. Also, our constructions include partial results about ASIC-POVMs constructed by X. Cao et al (X. Cao, J. Mi and S. Xu: Two constructions of approximately symmetric informationally complete positive operator-valued measures. J. Math. Phys., 58(6), 062201 (2017)). Mathematics Subject Classification (2020) 81P15 · 81P45

Subspaces coding plays an important role in error correction of random network coding. To study their properties and find good constructions, the notion of cyclic subspace codes was introduced using the extension field structure of the ambient space. Those cyclic constant-dimension subspace codes (CDCs) with optimal minimum distances may have additional structures that could be effectively applied in encoding and decoding algorithms (see Trautmann et al., IEEE Trans Inf Theory 59(11):7386–7404, 2013; Etzion and Vardy, IEEE Trans Inf Theory 57(2):1165–1173, 2011; Braun et al., Forum Math Pi 4(e7):1–14, 2016; Kohnert and Kurz, Math Methods Comput Sci 5393:31–42, 2008). In this paper, two new constructions of Sidon spaces are given by tactfully adding new parameters and flexibly varying the number of parameters. Under the parameters n= (2r+1)k, r ≥ 2 and p_0=max{i∈ℕ^+: ⌊r/i⌋ >⌊r/i+1⌋} , the first construction produces a cyclic CDC in 𝒢_q(n, k) with minimum distance 2k-2 and size ( (r+∑ _i=2^p_0(⌊r/i⌋ -⌊r/i+1⌋ ))(q^k-1)(q-1)+r) (q^k-1)^r-1(q^n-1)/q-1 . Given parameters n=2rk,r≥ 2 and if r=2 , p_0=1 , otherwise, p_0=max{ i∈ℕ^+: ⌈r/i⌉ -1>⌊r/i+1⌋} , a cyclic CDC in 𝒢_q(n, k) with minimum distance 2k-2 and size ( (r-1+∑ _i=2^p_0(⌈r/i⌉ -⌊r/i+1⌋ -1))(q^k-1)(q-1)+r-1) (q^k-1)^r-2⌊q^k-2/2⌋ (q^n-1)/q-1 is produced by the second construction. The sizes of our cyclic CDCs are larger than the best known results. In particular, in the case of n=4k , when k goes to infinity, the ratio between the size of our cyclic CDC and the Sphere-packing bound (Johnson bound) is approximately equal to 1/2 . Moreover, for a prime power q and positive integers k, s with 1≤ s< k-1 , a cyclic CDC in 𝒢_q(N, k) of size eq^N-1/q-1 and minimum distance ≥ 2k-2s is provided by subspace polynomials, where N, e are positive integers. Our construction generalizes previous results and, under certain parameters, provides cyclic CDCs with larger sizes or more admissible values of N than constructions based on trinomials.

Mutually unbiased bases (MUBs) are widely used in quantum information processing and play an important role in quantum cryptography, quantum state tomography and communications. It's diffi- ffi- cult to construct MUBs and remains unknown whether complete MUBs exist for any non prime power. Therefore, researchers have proposed the solution to construct approximately mutually unbiased bases (AMUBs) by weakening the inner product conditions. This paper constructs q AMUBs of Cq, q , (q q + 1) AMUBs of C q -1 and q AMUBs of C q -1 by using character sums over Galois rings and finite fields, where q is a power of a prime. The first construction of q AMUBs of Cq q is new which illustrates K AMUBs of CK K can be achieved. The second and third constructions in this paper include the partial results about AMUBs constructed by W. Wang et al. in [9].

Subspace codes, especially cyclic subspace codes, have attracted wide attention due to their applications in random network coding. In [14], Roth et al. presented the idea that cyclic subspace codes can be constructed employing Sidon spaces. In this paper, several kinds of Sidon spaces are first constructed and a cyclic subspace code with size \begin{document}$ l{q^k}\left( {\left\lceil {\frac{n}{{4k}}} \right\rceil - 1} \right)\frac{{{q^n} - 1}}{{q - 1}} $\end{document} and minimum distance \begin{document}$ 2k-2 $\end{document} is further given which improves and generalizes the previously known constructions, where \begin{document}$ n, k, l $\end{document} are positive integers, \begin{document}$ n $\end{document} is a multiple of \begin{document}$ k $\end{document} and \begin{document}$ l\leq k $\end{document}. Furthermore, in the case \begin{document}$ n = 3k, $\end{document} by considering the orbits of distinct Sidon spaces and the orbit of \begin{document}$ {\mathbb{F}_{{q^k}}}, $\end{document} a cyclic subspace code with size \begin{document}$ 2({q^k}-1) \frac{{{q^n} - 1}}{{q - 1}} + {q^{2k}}+{q^k} + 1 $\end{document} and minimum distance \begin{document}$ 2k-2 $\end{document} is obtained. As a consequence, we obtain more cyclic subspace codes with larger size of codewords than the previous works without decreasing the minimum distance.

Sidon space is an important tool for constructing cyclic subspace codes. In this letter, we construct some Sidon spaces by using primitive elements and the roots of some irreducible polynomials over finite fields. Let q be a prime power, k; m; n be three positive integers and rho = [m/2k] - 1, theta = [n/2m] -1. Based on these Sidon spaces and the union of some Sidon spaces, new cyclic subspace codes with size 3(q(n-)1)/q-1 and theta rho q(k) (q(n-)1)/q-1 are obtained. The size of these codes is lager compared to the known constructions from [14] and [10].

Codebooks with small maximum cross-correlation amplitudes are used to distinguish the signals from different users in CDMA communication systems. In this paper, we first study the Jacobi sums over Galois rings of arbitrary characteristics and completely determine their absolute values, which extends the work in [34], where the Jacobi sums over Galois rings with characteristics of a square of a prime number were discussed. Then, the deterministic construction of codebooks based on the Jacobi sums over Galois rings of arbitrary characteristics is presented, which produces asymptotically optimal codebooks with respect to the Welch bound. In addition, the parameters of the codebooks provided in this paper are new.

Let C be a constant dimension code whose codewords are k-dimensional subspaces of Fqm. One of the main research problems of constant dimension codes is to calculate the maximum possible cardinality Aq(m,d,k) of the code whose the distance between any two different codewords is at least d. In this paper, we propose a new construction approach of constant dimension codes. The constant dimension codes based on this construction can be inserted into the parallel linkage construction. Moreover, the proposed construction gives new lower bounds of Aq(m,d,k) for m=2k. Some constant dimension codes with larger cardinality than the previously best known codes are given.

利用有限域上典型群的几何空间构造子空间码,研究子空间码的基本问题是当前一项热门研究课题.基于伪辛空间的子空间构造子空间码,计算得到所构造的子空间码在最小距离确定的情况下码字个数最大值的不同上下界,分别是球填充界、Singleton界、Gilbert-Varshamov界和Wang-Xing-Safavi-Naini界,这些结果丰富了有限域上典型群的几何学研究内容.

Permutation polynomials over finite fields have been widely studied due to their important applications in mathematics and cryptography. In recent years, 2-to-1 mappings over finite fields were proposed to build almost perfect nonlinear functions, bent functions, and the semibent functions. In this paper, we generalize the 2-to-1 mappings to m-to-1 mappings, including their construction methods. Some applications of m-to-1 mappings are also discussed.

By using the important concept of stoping distance of the LDPC codes, how to perform about the iterative decoding of the LDPC codes in binary erasure channels is analyzed. In this article, we introduce a class of LDPC codes designed on the premise of the symplectic space over finite fields. An important parameter of stopping distance is estimated mainly and the lower bound on the stopping distance of the code designed on the premise of symplectic space is acquired.

In this article, we compute the Sphere-packing bound, Singleton bound, Gilbert-Varshamov bound and Wang-Xing-Safavi-Naini (W-X-S-N) bound on the subspace codes (2n+σ,H,ρ,(s,0,0,0))q on the premise of subspaces with the form of (s,0,0,0) in pseudo-sympletic spaces Fq2n+σ and on the subspace codes (2n+σ+μ,H,ρ,(s,0,0,0))q on the premise of subspaces with the form of (s,0,0,0) in singular pseudo-sympletic spaces Fq(2n+σ+μ).

Subspace codes are widely used in error corrections of random network coding. In this article, subspace codes based on partial injective maps of vector spaces over finite fields are considered. Several bounds of the subspace codes (n, M, 2b, e)q based on e-partial injective maps of F(n)q are presented. The anticode bound and Ahlswede-Aydinian bound of the subspace codes (n, M, 2b, e)q are obtained by using the EKR theorem for e-partial injective maps of F(n)q . Finally, we show that the (n, M, 2b, e)q subspace codes based on e-partial injective maps of F(n)q reach the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures in Ine.

In this paper, we construct some 11/2-designs, which are also known as partial geometric designs, via singular linear space. Furthermore, we will give four families of directed strongly regular graphs by using those 11/2-designs.

Recently, Luo et al. proposed two new codebooks based on the operations of some given sets, which are asymptotically optimal. In the paper, we present new construction of codebooks. We determine maximal cross-correlation amplitude of the constructed codebooks. The constructed codebooks are near optimal to the Welch bound. This new codebooks in the paper generalize Luo et al.'s constructions. The parameters of constructed codebooks are nimble and new under some circumstances. (C) 2020 Elsevier Inc. All rights reserved.

In this paper, we study some bounds of constant dimension codes further in Grassmannian space G(q)(n, k). There is an increasing interest in subspace codes since they are precisely what is needed for errors-correction in networks. There is also a connection to the theory over finite fields. By revising the specific construction method of the constant dimension codes in [1], [2], we can improve some bounds on q-ary constant dimension codes in some given cases.

In this letter, motivated by the research of Tian et al., two constructions of asymptotically optimal codebooks in regard to the Welch bound with additive and multiplicative characters are provided. The parameters of constructed codebooks are new, which are di fferent from those in the letter of Tian et al.

The group testing problem is that we are asked to identify all the defects with the minimum number of tests when given a set of n items with at most d defects. In this paper, as a generalization of Liu et al.’s construction in the paper (Liu and Gao in Discret Math 338:857–862, 2015), new pooling designs are constructed from singular linear spaces over finite fields. Then we make comparisons with Liu et al.’s construction in the aspects of parameters of pooling designs. By choosing appropriate parameters in our pooling designs, the performance of test efficiency in our pooling designs is better than that given by Liu et al. Finally, the analysis of parameters in our pooling designs is provided.

Compressed sensing (CS) is a new data acquisition theory taking full use of the sparsity of signals. It reveals that higher-dimensional sparse signals can be reconstructed from fewer nonadaptive linear measurements. The construction of CS matrices in CS is the key problem. In this paper, the deterministic CS matrices from optimal codebooks are constructed. Furthermore, the maximum sparsity of recovering the sparse signals by using our CS matrices are obtained. Meanwhile, a comparison is made with the CS matrices constructed by DeVore based on polynomials over finite fields. In the numerical simulations, our CS matrix outperforms DeVore[Formula: see text]s matrix in the process of recovering sparse signals.

In a number of applications, the codebook that minimizes the maximal cross-correlation amplitude (Imax) is often desirable. There are two major ingredients in this paper. These two contents are to propose two constructions of codebooks. The codebooks generated by these constructions asymptotically meet the Welch bound. What we find is the codebook constructed in Section 2 can also asymptotically meet the Levenshtein bound. The parameters of these codebooks are new.