.. _sphx_glr_auto_examples_coding:

Coding and shaping
------------------

Channel codes against the finite-length limits and the literature, the codes of 5G NR,
probabilistic amplitude shaping.


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    <div class="sphx-glr-thumbcontainer" tooltip="BPSK over AWGN, decoded from channel LLRs 2y/\sigma^2.">

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  .. image:: /auto_examples/coding/images/thumb/sphx_glr_plot_01_channel_coding_thumb.png
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  :doc:`/auto_examples/coding/plot_01_channel_coding`

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      <div class="sphx-glr-thumbnail-title">Channel coding: from convolutional codes to the finite-length limit</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Fig. 5 of X.-Y. Hu, E. Eleftheriou and D. M. Arnold, &quot;Regular and irregular progressive edge-growth Tanner graphs&quot;, IEEE Trans. Inf. Theory 51(1), 2005, gives the bit and block error rates of a regular LDPC code built by progressive edge growth, with n = 504, m = 252 parity checks, symbol-node degree d_s = 3 and check-node degree d_c = 6, over the AWGN channel with BPSK and sum-product decoding, next to MacKay&#x27;s code and a random graph of the same parameters.">

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  .. image:: /auto_examples/coding/images/thumb/sphx_glr_plot_02_hu2005_fig5_thumb.png
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  :doc:`/auto_examples/coding/plot_02_hu2005_fig5`

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      <div class="sphx-glr-thumbnail-title">Hu, Eleftheriou & Arnold 2005, Fig. 5: a regular PEG code of length 504</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="The data channels of 5G NR use a quasi-cyclic LDPC code, the control channels a polar code (3GPP TS 38.212; T. Richardson and S. Kudekar, &quot;Design of low-density parity check codes for 5G new radio,&quot; IEEE Commun. Mag. 56(3), 2018; V. Bioglio, C. Condo and I. Land, &quot;Design of polar codes in 5G New Radio,&quot; IEEE Commun. Surveys Tuts. 23(1), 2021). comnum.NrLdpc and comnum.NrPolar build them from the tables of the standard — the two LDPC base graphs, the polar sequence of 1024 positions, the interleavers — with their rate matching to any number of transmitted bits.">

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  .. image:: /auto_examples/coding/images/thumb/sphx_glr_plot_03_nr_codes_thumb.png
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  :doc:`/auto_examples/coding/plot_03_nr_codes`

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      <div class="sphx-glr-thumbnail-title">5G NR channel codes: LDPC and polar</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Rate. A uniform QAM wastes energy on its outer points: at a given SNR its rate stays below the capacity \log_2(1 + \mathrm{SNR}) of the AWGN channel, by up to the 1.53 dB of the ultimate shaping gain. Maxwell-Boltzmann probabilities p(x) \propto e^{-\lambda |x|^2} (Kschischang &amp; Pasupathy 1993) close most of that gap: the curve is the GMI of bit-metric decoding of 64-QAM with the priors, H(X) - \sum_k H(B_k \mid Y) (Böcherer 2014), with the best \lambda at every SNR, and the points are the estimates of simulated links (``specs/shaping_awgn.yaml`` checks them against the closed forms).">

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  .. image:: /auto_examples/coding/images/thumb/sphx_glr_plot_04_probabilistic_shaping_thumb.png
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  :doc:`/auto_examples/coding/plot_04_probabilistic_shaping`

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      <div class="sphx-glr-thumbnail-title">Probabilistic shaping: the rate gain and probabilistic amplitude shaping</div>
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.. toctree::
   :hidden:

   /auto_examples/coding/plot_01_channel_coding
   /auto_examples/coding/plot_02_hu2005_fig5
   /auto_examples/coding/plot_03_nr_codes
   /auto_examples/coding/plot_04_probabilistic_shaping