Polar code with block-length N = 3n

The generator matrix of the polar codes proposed by Arikan takes a form of G^⊗n₂, where G₂ is a 2 × 2 kernel matrix, and ^⊗n denotes the Kronecker product [1]. However we can also construct polar codes with other specific G_l, 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^⊗n₃ and 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, G₃ is 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.