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• W3115431353 abstract "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 $epsilon > 2epsilon^*$, where $epsilon^* approx 0.085$, an existence of sequence of capacity-achieving codes with all the column wights of the generator matrix upper bounded by $(log N)^{1+epsilon}$ 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 $lambda >lambda^* = 0.5+epsilon^*$, the existence of a sequence of capacity-achieving codes where all the column wights of the generator matrix are bounded from above by $(log N)^{2lambda}$ and with decoding complexity $O(Nlog log N)$ is shown. The existence of similar capacity-achieving LDGM codes with low-complexity decoding is shown for any BMS channel, and for any $lambda >lambda^{dagger} approx 0.631$." @default.
• W3115431353 created "2021-01-05" @default.
• W3115431353 creator A5056299839 @default.
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• W3115431353 date "2020-12-27" @default.
• W3115431353 modified "2023-09-27" @default.
• W3115431353 title "Capacity-achieving Polar-based LDGM Codes." @default.
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