The Weight Enumerator of Three Families of Cyclic Codes

Cyclic codes are a subclass of linear codes and have wide applications in consumer electronics, data storage systems, and communication systems due to their efficient encoding and decoding algorithms. Cyclic codes with many zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a very small number of cases. In general the calculation of the weight distribution of cyclic codes is heavily based on the evaluation of some exponential sums over finite fields. Very recently, Li, Hu, Feng and Ge studied a class of $p$-ary cyclic codes of length $p^{2m}-1$, where $p$ is a prime and $m$ is odd. They determined the weight distribution of this class of cyclic codes by establishing a connection between the involved exponential sums with the spectrum of Hermitian forms graphs. In this paper, this class of $p$-ary cyclic codes is generalized and the weight distribution of the generalized cyclic codes is settled for both even $m$ and odd $m$ alone with the idea of Li, Hu, Feng, and Ge. The weight distributions of two related families of cyclic codes are also determined.Comment: 13 Pages, 3 Table

A Family of Five-Weight Cyclic Codes and Their Weight Enumerators

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a family of $p$-ary cyclic codes whose duals have three zeros are proposed. The weight distribution of this family of cyclic codes is determined. It turns out that the proposed cyclic codes have five nonzero weights.Comment: 14 Page

Normal Bases on Galois Ring Extensions

Normal bases are widely used in applications of Galois fields and Galois rings in areas such as coding, encryption symmetric algorithms (block cipher), signal processing, and so on. In this paper, we study the normal bases for Galois ring extension R / Z p r , where R = GR ( p r , n ) . We present a criterion on the normal basis for R / Z p r and reduce this problem to one of finite field extension R ¯ / Z ¯ p r = F q / F p ( q = p n ) by Theorem 1. We determine all optimal normal bases for Galois ring extension

New Construction of Maximum Distance Separable (MDS) Self-Dual Codes over Finite Fields

Maximum distance separable (MDS) self-dual codes have useful properties due to their optimality with respect to the Singleton bound and its self-duality. MDS self-dual codes are completely determined by the length n , so the problem of constructing q-ary MDS self-dual codes with various lengths is a very interesting topic. Recently X. Fang et al. using a method given in previous research, where several classes of new MDS self-dual codes were constructed through (extended) generalized Reed-Solomon codes, in this paper, based on the method given in we achieve several classes of MDS self-dual codes

MiR-361-5p promotes proliferation and inhibits apoptosis of fibroblast-like synoviocytes via targeting ZBTB10 in rheumatoid arthritis

Objectives This study is aimed to explore the key role of miR-361-5p in fibroblast-like synovial (FLS) cells of rheumatoid arthritis (RA) and explore the underlying mechanism. Methods First, we performed RT-qPCR to evaluate the expression of miR-361-5p in both synovial tissues of RA patients and cultured RA-FLS cells. Then CCK-8 assay, EdU staining, Western blot, flow cytometry, and ELISA were conducted to estimate the influence of inhibiting miR-361-5p on RA-FLS cells. Moreover, we used bioinformatics analysis to predict the potential targets of miR-361-5p and perform a dual luciferase report assay for verification. Finally, rescue experiments were performed to prove the role of miR-361-5p/Zinc Finger And BTB Domain Containing 10 (ZBTB10) in the proliferation, cell cycle, and apoptosis of RA-FLS. Results We find that the expression of miR-361-5p is increased in both RA tissues and cultured RA-FLS cells. The inhibition of miR-361-5p can not only inhibit proliferation, arrest the cell cycle in G1/G0 phase, and increase apoptosis, but also reduce the inflammatory factors secreted by RA-FLS cells. In addition, ZBTB10 is a direct target for miR-361-5p, over-expression of ZBTB10 reverses the effect of miR-361-5p in RA-FLS. Conclusions MiR-361-5p promotes the progression of rheumatoid arthritis by targeting ZBTB10.Key points The influences of miR-361-5p on RA-FLS cells