In this paper, we study the weight spectrum of linear codes with super-linear field size and use probabilistic method to show that for nearly all such codes, corresponding is very close a maximum distance separable (MDS) code.

We prove that, for the binary erasure channel (BEC), polar-coding paradigm gives rise to codes that not only approach Shannon limit but do so under best possible scaling of their block length as a function gap capacity. This result exhibits first known family attain both optimal and quasi-linear complexity encoding decoding. Our proof is based on construction analysis polar with large kernels. ...