Extended Polar Codes Perform Better in Terms of Compound Rate and Scaling Behavior

Polar codes provably achieve the capacity of a wide array of channels under successive decoding. However, in terms of some practical aspects, these codes do not perform optimally and need further improvement. In this paper, we consider two such aspects: universality and finite-length scaling. We argue analytically that for extensions of polar codes that are based on l x l matrices, these aspects become optimal as l grows large.