Consider a simple, connected graph Γ withnvertices. LetCbe a code of lengthnwith its coordinates corresponding to the vertices of Γ. We defineCas astorage codeon Γ if, for any codewordc∈C, the information at each coordinate ofccan be recovered by accessing its neighboring coordinates. The main problem here is to construct high-rate storage codes on triangle-free graphs. In this paper, we employ the polynomial method to address a question proposed by Barg and Zémor in 2022, demonstrating that the BCH family of storage codes on triangle-free Cayley graphs achieves a unit rate. Furthermore, we generalize the construction of the BCH family and obtain more storage codes of unit rate on triangle-free graphs. We also compare the BCH family with the other known constructions by examining the rate of convergence of 1/(1-R(Cn)) with respect to the lengthn, whereR(Cn) is the rate of codeCn. At last, we reveal a connection between the storage codes on triangle-free graphs and the Ramsey numberR(3,t), which leads to an upper bound for the rate of convergence of 1/(1 -R(Cn)).

Solving the large-scale linear system of equations is one of the most fundamental problems both in theory and practice. However this problem requires too much computational resource for most users to solve it. With the rapid development of cloud services, many users tend to outsource the expensive computing to the cloud server, which is regarded as an efficient way of solving such problem. Nevertheless, the cloud server can not protect the data privacy well, especially when the user's linear system of equations contain private and sensitive data. There are many previous research works on secure outsourcing of systems of linear equations. In this paper we first analyze a privacy preserving CGM (conjugate gradient method) algorithm for secure outsourcing of large-scale systems of linear equations proposed in [1] . We find that the cloud server can recover the protected coefficient matrix of the linear system of equations from the message it receives, which makes the security method in this scheme fails. This is a serious problem, which makes the private and sensitive data of the user leak to the cloud server, and privacy preserving does not exist. To overcome this problem, we modified this algorithm to protect the message from leaking, which can protect the users' privacy well. We also show the security of this new scheme and do experiments to show its efficiency.

Sequence set with lower correlation values is highly desired for engineering applications. However, theoretical results (e.g., Welch bound) show that θmax≥ √(N) in general, that is, the maximum out-of-phase autocorrelation and cross-correlation magnitudes of a sequence set is not less than the square root of the sequence period. In this paper, we propose a new concept, namely almost perfect sequence set (APSS), which has the property θmax≤ c except for at most m shifts, where c and m are predefined small integers. A uniform method is presented to construct APSS and then the properties of such APSS are discussed. Moreover, a distance inequality on the APSS with m = 1 is obtained and several APSS families such as (2p, 8p + 2, 6, 4) -APSS and (3p, (64p2+8)/3,9,9) -APSS for any prime p ≥ 5 are constructed based on Paley and Paley partial sequences. Finally, it shows that the APSS can be used to construct LCZ sequences and the properties of such LCZ sequences are presented.

Permutation polynomials have applications in coding theory, cryptography, combinatorial designs, and they have been studied for over a hundred years. Recently, permutation polynomials over Znhave been used to construct interleavers for turbo codes. In this paper, we determine all permutation polynomials over Znwith degree no more than six.