ary erasure channel, the Bhattacharyya parameter can be calculated using recursive formulas. Thus the polar code construction problem can be solved without a need for approximation. The Bhattacharyya parameter can be calculated using the following recursive formulas with a complexity of O(N) [29]: For the AWGN channel no efficient algorithm for calculating the Bhattacharyya parame- ter per synthesized channel is known. Approximating the exact polar code construction is possible to reduce complexity by calculating an estimate of the Bhattacharyya parameter per synthesized channel. Several suboptimal construction methods were proposed in the literature with different computational complexities. The main difference between polar codes and Reed-Muller codes is the choice of the information set A. In the case of Reed-Muller codes, the indices of the highest weight rows of the generator matrix G are selected to carry the information. A polar code of length N = 2n is generated using genera- tor matrix G of size NXN. A block of length N, consisting of N - K frozen bits and K information bits, is multiplied by G to produce the polar codeword X = uG. The generator matrix G is based on a kernel that is used to construct the code, where which x denotes the Kronecker product.11 A polar encoding lattice, equivalent to G, can also be used as a polar encoder, as shown in Fig. 4.45. Note that successive graph represen- tations have recursive relationships. More specifically, the graph representation for a polar encoding kernel operation having a kernel block size of N = 2 comprises a single stage, containing a single XOR. The first of the N = 2 kernel encoded bits is obtained as the XOR of the N = 2 kernel information bits, while the second kernel encoded bit is equal to the second kernel information bit. For greater kernel block sizes N, the graph representation may be considered to be a vertical concatenation of two graph representations for a kernel block size of N = 2, followed by an additional stage of XORs, as shown in Fig.