Coded caching is a promising technique to effectively reduce peak traffic by using local caches and the multicast gains generated by these local caches. We prefer to design a coded caching scheme with the subpacketization $F$ and transmission load $R$ as small as possible since these are the key metrics for evaluating the implementation complexity and transmission efficiency of the scheme, respectively. However, most of the existing coded caching schemes have large subpacketizations which grow exponentially with the number of users $K$, and there are a few schemes with linear subpacketizations which have large transmission loads. In this paper, we focus on studying the linear subpacketization, i.e., $K=F$, coded caching scheme with low transmission load. Specifically, we first introduce a new combinatorial structure called non-half-sum disjoint packing (NHSDP) which can be used to generate a coded caching scheme with $K=F$. Then a class of new schemes is obtained by constructing NHSDP. Theoretical and numerical comparisons show that (i) compared to the existing schemes with linear subpacketization (to the number of users), the proposed scheme achieves a lower load; (ii) compared to some existing schemes with polynomial subpacketization, the proposed scheme can also achieve a lower load in some cases; (iii) compared to some existing schemes with exponential subpacketization, the proposed scheme has loads close to those of these schemes in some cases. Moreover, the new concept of NHSDP is closely related to the classical combinatorial structures such as cyclic difference packing (CDP), non-three-term arithmetic progressions (NTAP), and perfect hash family (PHF). These connections indicate that NHSDP is an important combinatorial structure in the field of combinatorial design.