#### Scattered EXIT Charts for Finite Length LDPC Code Design

##### Moustafa Ebada, Ahmed Elkelesh, Sebastian Cammerer, Stephan ten Brink

We introduce the Scattered Extrinsic Information Transfer (S-EXIT) chart as a tool for optimizing degree profiles of short length Low-Density Parity-Check (LDPC) codes under iterative decoding. As degree profile optimization is typically done in the asymptotic length regime, there is space for further improvement when considering the finite length behavior. We propose to consider the average extrinsic information as a random variable, exploiting its specific distribution properties for guiding code design. We explain, step-by-step, how to generate an S-EXIT chart for short-length LDPC codes. We show that this approach achieves gains in terms of bit error rate (BER) of 0.5 dB and 0.6 dB over the additive white Gaussian noise (AWGN) channel for codeword lengths of 128 and 180 bits, respectively, at a target BER of $10^{-4}$ when compared to conventional Extrinsic Information Transfer (EXIT) chart-based optimization. Also, a performance gain for the Binary Erasure Channel (BEC) for a block (i.e., codeword) length of 180 bits is shown.

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