Capacity-achieving Polar-based LDGM Codes
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In this paper, we study codes with sparse generator matrices. More specifically, low-density generator matrix (LDGM) codes with a certain constraint on the weight of all the columns in the generator matrix are considered. In this paper, it is first shown that when a binary-input memoryless symmetric (BMS) channel W and a constant s>0 are given, there exists a polarization kernel such that the corresponding polar code is capacity-achieving and the column weights of the generator matrices are bounded from above by N^s. Then, a general construction based on a concatenation of polar codes and a rate-1 code, and a new column-splitting algorithm that guarantees a much sparser generator matrix is given. More specifically, for any BMS channel and any ϵ > 2ϵ^*, where ϵ^* ≈ 0.085, an existence of sequence of capacity-achieving codes with all the column wights of the generator matrix upper bounded by (log N)^1+ϵ is shown. Furthermore, coding schemes for BEC and BMS channels, based on a second column-splitting algorithm are devised with low-complexity decoding that uses successive-cancellation. The second splitting algorithm allows for the use of a low-complexity decoder by preserving the reliability of the bit-channels observed by the source bits, and by increasing the code block length. In particular, for any BEC and any λ >λ^* = 0.5+ϵ^*, the existence of a sequence of capacity-achieving codes where all the column wights of the generator matrix are bounded from above by (log N)^2λ and with decoding complexity O(Nloglog N) is shown. The existence of similar capacity-achieving LDGM codes with low-complexity decoding is shown for any BMS channel, and for any λ >λ^†≈ 0.631.
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