Polar code with block-length N = 3n
Liang Zhang, Zhaoyang Zhang, Xianbin Wang · 2012
The generator matrix of the polar codes proposed by Arikan takes a form of G⊗n2, where G2is a 2 × 2 kernel matrix, and⊗ndenotes the Kronecker product [1]. However we can also construct polar codes with other specific Gl, where l is an arbitrary integer and l ≥ 2. This paper considers the construction and decoding of a class of polar codes with a generator matrix of G⊗n3and a 3 × 3 kernel matrix of G3. Unlike the 2 × 2 one that has only a single linear form, the 3×3 kernel matrix can take different forms, which needs more selectivity in the code construction. We thus give some design criteria to find the best G3, and then show how to construct and decode the polar code in detail. Encoding and decoding complexity are analyzed. It should be pointed out that, G3is the simplest case which can be used to illustrate the construction of a polar code among different choices of kernel matrices, and such idea can also be generalized to the construction of polar codes with other l.