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Known results are very limited, and a lot of them are based on the works of Denniston \\cite{MR0349416, MR0369086, MR535159, MR539718}. The only known recursive constructions are an tripling construction by Denniston \\cite{MR535159}and a product construction by Lei \\cite{MR1931492}, both constructs an LKTS($uv$) on the basis of an LKTS($v$).\n  In this paper, we describe an construction of LKTS$(q^n+2)$ from LKTS$(q+2)$, where $q$ is a prime power of the form $6t+1$. 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