Sparse arrays with $N$-sensors can provide up to $O(N^2)$ degrees of freedom (DOF) by second-order cumulants. However, these sparse arrays like minimum-/low-redundancy arrays (MRAs/LRAs), nested arrays and coprime arrays can only provide limited DOF and array aperture with the same number of physical sensors. However, further increasing DOF would increase costs in practical applications. The paper aims to design a sparse linear array (SLA) with higher DOF and lower redundancy via exploring different cases of third-order cumulants. Based on the framework third-order exhaustive co-array (TO-ECA), a general third-order array (GTOA) with any generator is proposed in the paper. Further, three novel arrays are designed based on GTOA with different generators, namely third-order sum and difference array (generator) (TO-SDA(CNA)), (TO-SDA(SCNA)) and (TO-SDA(TNA-II)) which can provide closed-form expressions for the sensor locations and enhance DOF in order to resolve more signal sources in the estimation of direction of arrival (DOA). The three TO-SDAs are all consisted of two sub-arrays, where the first is the generator and another is a ULA with big inter-spacing between sensors. For the three TO-SDAs, the maximum DOF under the given number of total physical sensors can be derived and the TO-ECA of the three TO-SDAs are hole-free. Additionally, the redundancy of the three TO-SDAs is defined, and the lower band of the redundancy for the three TO-SDAs is derived. Furthermore, the proposed TO-SDA(TNA-II) not only achieves higher DOF than those of existing TONA and even SE-FL-NA but also reduces mutual coupling effects. Meanwhile it realizes higher resolution and decreases redundancy. Numerical simulations are conducted to verify the superiority of TO-SDA(TNA-II) on DOA estimation performance and enhanced DOF over other existing DCAs.