 the form of xH for some XE( is said to be a left coset of H and a subset of the form Hx is said to be a right coset of H. For any subgroup H, we can define an equivalence relation by x~y if x = yh for some heH. The equivalence classes of this equivalence relation are exactly the left cosets of H, and an element X of H is in the equivalence class xH. Thus the left cosets of H form a partition of G. It is also true that any two left cosets of H have the same cardinal number, and in particular, every coset of H has the same cardinal number as eH = H, where e is the identity element. Thus the cardinal num- ber of any left coset of H has cardinal number the order of H. The same results are true of the right cosets of G and, in fact, one can prove that the set of left cosets of H has the same cardinal number as the set of right cosets of H. 518 Chapter 4 potential for list decoding, because of its sequential hierarchical structure. It was proved that polar codes achieve Shannon capacity of any symmetric binary-input discrete memory- less channel under SC decoding [33,34]. SCL decoder was proposed as an extended version of SC decoder where instead of succes- sively computing hard decisions for each bit, it branches one SC decoder into two parallel SC-decoders at each stage of decision where each branch has its path metric that is continu- ously updated for each path. It can be shown that a list of size 32 is enough to almost achieve the ML bound. SCL with CRC decoder is an extension of SCL decoder, where a high-rate CRC code is appended to the polar code, SO that the correct codeword is selected among the candidate codewords from the final list of paths. It was observed that whenever an SCL-decoder fails, the correct codeword exists in the list. Therefore, the CRC was proposed as a validity check for each candidate codeword in the list. In BP decoder, unlike SC-based decoding techniques, there is no inter-bit dependence and thus no error propagation. It does not encounter any intermediate hard decisions. It updates the