Binary sequences with optimal autocorrelation and large linear complexity have important applications in cryptography and communications. Very recently, a class of binary sequences of period $4p$ with optimal autocorrelation was proposed via interleaving four suitable Ding-Helleseth-Lam sequences (Des. Codes Cryptogr., DOI 10.1007/s10623-017-0398-5), where $p$ is an odd prime with $p\equiv 1(\bmod~4)$. The objective of this paper is to determine the minimal polynomial and the linear complexity of this class of binary optimal sequences via sequence polynomial approach. It turns out that this class of sequences has quite good linear complexity.