Multi-kernel polar codes, which are constructed by different sizes of kernels make the traditional polar codes have more flexible code length. In this paper, we extend the multi-kernel polar codes to 5x5 kernels including kernel design, code design, and fast decoding. First, we construct a new 5x5 kernel, which has the maximum polarization rate. The first row of this kernel is all-one vector, which enable us to seek the relation between node modification and distance property of multi-kernel polar codes. Second, we modify the repetition (REP) nodes to this kernel, which can increase the distance property of the constructed polar codes, improve the decoding performance and reduce memory requirements during the fast decoding. In addition, we find this REP node modification can be generalized when multi-kernel polar codes constructed by kernels, which the first row is all-one vector. Third, we prove the four kinds of basic nodes (Rate-1 node, Rate-0 nodes, Repetition (REP) nodes, and Single-Parity check (SPC) nodes) of fast decoding are suitable for this kernel. Moreover, we generalize the Generalized Parity-Check (G-PC) nodes and the Generalized Repetition (G-REP) nodes, which are suitable for multi-kernel polar codes constructed by any size of kernels and any order of kernels. The G-PC and G-REP nodes can further accelerate the decoding speed. By simulations, we show that our modification improves the error-rate performance of the fast decoding. Meanwhile, the fast decoding of our multi-kernel polar codes has at least 68.4% reduced latency compared with the traditional SC decoder in all cases considered where frame lengths are (80, 135, 400, 675).
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