Linear recurring sequences and weight distributions of some cyclic codes

In this paper, a family of 2r-dimensional cyclic codes is considered. In some cases, the weight distribution of these codes is determined. For this family of cyclic codes, the upper and lower bounds on the number of non-zero weights that the codes can have are obtained. In addition, it is proved in Shi et al. (Chin J Electron 28(6):1127-113, 2019) that any 2-dimensional cyclic code has at most two non-zero weights, and an open problem is proposed to extend this result to higher-dimensional cyclic codes. This paper also partially solves the open problem. As an application, we construct two families of strongly regular graphs employing 2-weight projective codes and calculate their parameters, eigenvalues, and multiplicities. One of them has the parameters of the Lattice graphs, and the other family has the parameters of the Latin Square graphs.