An Analysis of the Properties of Polar Codes

Polar Codes have risen to the forefront of practical coding theory, due to their incredible efficiency and ease of construction. Originally introduced by Arikan in his 2009 paper, they are the first code defined with an explicit construction that achieved channel capacity. Moreover, polar codes possess some physically practical properties that make their implementation alluring. In the same paper as mentioned above, Arikan elaborated on his construction and noted that the construction given was one specific instance of a polar code and that there is a family of polar codes that can be produced by the same method. Since this method of construction relies on repeated application of the Kronecker product, a series of questions surrounding how the Kronecker product interacts with the properties of these polar codes have begun to arise. First, we will focus on producing a handful of fundamental lemmas and results regarding the interaction of the Kronecker product concerning the Hamming metric and the Hamming weight. This will allow us to navigate the successive application of the tensor product more easily. Second, we will apply these fundamental lemmas to the original construction of Arikan to show that the properties of the polar code produced can be determined from the starting matrix. From there, we will attempt to abstract the property of polar codes to a more general setting. Lastly, we will conclude with a series of notes explaining the difficulty of generalizing this result to arbitrary matrices.