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<title>Differential pulse-code modulation</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Differential pulse-code modulation</span></span>
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<p><b>Differential pulse-code modulation</b> (<b>DPCM</b>) is a signal encoder that uses the baseline of <a href="Pulse-code_modulation" title="Pulse-code modulation">pulse-code modulation</a> (PCM) but adds some functionalities based on the prediction of the samples of the signal. The input can be an <a href="Analog_signal" title="Analog signal">analog signal</a> or a <a href="Digital_signal_(signal_processing)" title="Digital signal (signal processing)">digital signal</a>.
</p><p>If the input is a <a href="Continuous-time" class="mw-redirect" title="Continuous-time">continuous-time</a> analog signal, it needs to be <a href="Sampling_(signal_processing)" title="Sampling (signal processing)">sampled</a> first so that a <a href="Discrete-time_signal" class="mw-redirect" title="Discrete-time signal">discrete-time signal</a> is the input to the DPCM encoder.
</p>
<ul><li>Option 1: take the values of two consecutive samples; if they are analog samples, <a href="Quantization_(signal_processing)" title="Quantization (signal processing)">quantize</a> them; calculate the difference between the first one and the next; the output is the difference.</li>
<li>Option 2: instead of taking a difference relative to a previous input sample, take the difference relative to the output of a local model of the decoder process; in this option, the difference can be quantized, which allows a good way to incorporate a controlled loss in the encoding.</li></ul>
<p>Applying one of these two processes, short-term redundancy (positive correlation of nearby values) of the signal is eliminated; compression ratios on the order of 2 to 4 can be achieved if differences are subsequently <a href="Entropy_coded" class="mw-redirect" title="Entropy coded">entropy coded</a> because the entropy of the difference signal is much smaller than that of the original discrete signal treated as independent samples.
</p><p>DPCM was invented by <a href="C._Chapin_Cutler" title="C. Chapin Cutler">C. Chapin Cutler</a> at <a href="Bell_Labs" title="Bell Labs">Bell Labs</a> in 1950; his patent includes both methods.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
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<div class="mw-heading mw-heading2"><h2 id="Option_1:_difference_between_two_consecutive_quantized_samples">Option 1: difference between two consecutive quantized samples</h2></div>
<p>The encoder performs the function of differentiation; a quantizer precedes the differencing of adjacent quantized samples; the decoder is an accumulator, which if correctly initialized exactly recovers the quantized signal.
</p>
<div class="mw-heading mw-heading2"><h2 id="Option_2:_analysis_by_synthesis">Option 2: analysis by synthesis</h2></div>
<p>The incorporation of the decoder inside the encoder allows quantization of the differences, including nonlinear quantization, in the encoder, as long as an approximate inverse quantizer is used appropriately in the receiver. When the quantizer is uniform, the decoder regenerates the differences implicitly, as in this simple diagram that Cutler showed:
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Adaptive_differential_pulse-code_modulation" title="Adaptive differential pulse-code modulation">Adaptive differential pulse-code modulation</a></li>
<li><a href="Delta_modulation" title="Delta modulation">Delta modulation</a>, a special case of DPCM where the differences e<sub>Q</sub>[n] are represented with 1 bit as ±Δ</li>
<li><a href="Pulse_modulation_methods" class="mw-redirect" title="Pulse modulation methods">Pulse modulation methods</a></li>
<li><a href="Delta-sigma_modulation" title="Delta-sigma modulation">Delta-sigma modulation</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">U.S. patent 2605361, C. Chapin Cutler, <a rel="nofollow" class="external text" href="https://patents.google.com/patent/US2605361">"Differential Quantization of Communication Signals"</a>, filed June 29, 1950, issued July 29, 1952</span>
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</style><div id="Data_compression_methods241" style="font-size:114%;margin:0 4em"><a href="Data_compression" title="Data compression">Data compression</a> methods</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Lossless_compression" title="Lossless compression">Lossless</a><br>type</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Entropy_coding" title="Entropy coding">Entropy</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Adaptive_coding" title="Adaptive coding">Adaptive coding</a></li>
<li><a href="Arithmetic_coding" title="Arithmetic coding">Arithmetic</a></li>
<li><a href="Asymmetric_numeral_systems" title="Asymmetric numeral systems">Asymmetric numeral systems</a></li>
<li><a href="Golomb_coding" title="Golomb coding">Golomb</a></li>
<li><a href="Huffman_coding" title="Huffman coding">Huffman</a>
<ul><li><a href="Adaptive_Huffman_coding" title="Adaptive Huffman coding">Adaptive</a></li>
<li><a href="Canonical_Huffman_code" title="Canonical Huffman code">Canonical</a></li>
<li><a href="Modified_Huffman_coding" title="Modified Huffman coding">Modified</a></li></ul></li>
<li><a href="Range_coding" title="Range coding">Range</a></li>
<li><a href="Shannon_coding" title="Shannon coding">Shannon</a></li>
<li><a href="Shannon%E2%80%93Fano_coding" title="Shannon–Fano coding">Shannon–Fano</a></li>
<li><a href="Shannon%E2%80%93Fano%E2%80%93Elias_coding" title="Shannon–Fano–Elias coding">Shannon–Fano–Elias</a></li>
<li><a href="Tunstall_coding" title="Tunstall coding">Tunstall</a></li>
<li><a href="Unary_coding" title="Unary coding">Unary</a></li>
<li><a href="Universal_code_(data_compression)" title="Universal code (data compression)">Universal</a>
<ul><li><a href="Exponential-Golomb_coding" title="Exponential-Golomb coding">Exp-Golomb</a></li>
<li><a href="Fibonacci_coding" title="Fibonacci coding">Fibonacci</a></li>
<li><a href="Elias_gamma_coding" title="Elias gamma coding">Gamma</a></li>
<li><a href="Levenshtein_coding" title="Levenshtein coding">Levenshtein</a></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Dictionary_coder" title="Dictionary coder">Dictionary</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Byte-pair_encoding" title="Byte-pair encoding">Byte-pair encoding</a></li>
<li><a href="LZ77_and_LZ78" title="LZ77 and LZ78">Lempel–Ziv</a>
<ul><li><a href="842_(compression_algorithm)" title="842 (compression algorithm)">842</a></li>
<li><a href="LZ4_(compression_algorithm)" title="LZ4 (compression algorithm)">LZ4</a></li>
<li><a href="LZJB" class="mw-redirect" title="LZJB">LZJB</a></li>
<li><a href="Lempel%E2%80%93Ziv%E2%80%93Oberhumer" title="Lempel–Ziv–Oberhumer">LZO</a></li>
<li><a href="LZRW" title="LZRW">LZRW</a></li>
<li><a href="Lempel%E2%80%93Ziv%E2%80%93Storer%E2%80%93Szymanski" title="Lempel–Ziv–Storer–Szymanski">LZSS</a></li>
<li><a href="Lempel%E2%80%93Ziv%E2%80%93Welch" title="Lempel–Ziv–Welch">LZW</a></li>
<li><a href="LZWL" title="LZWL">LZWL</a></li>
<li><a href="Snappy_(compression)" title="Snappy (compression)">Snappy</a></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Burrows%E2%80%93Wheeler_transform" title="Burrows–Wheeler transform">BWT</a></li>
<li><a href="Context_tree_weighting" title="Context tree weighting">CTW</a></li>
<li><a href="Context_mixing" title="Context mixing">CM</a></li>
<li><a href="Delta_encoding" title="Delta encoding">Delta</a>
<ul><li><a href="Incremental_encoding" title="Incremental encoding">Incremental</a></li></ul></li>
<li><a href="Dynamic_Markov_compression" title="Dynamic Markov compression">DMC</a></li>
<li><a href="Grammar-based_code" title="Grammar-based code">Grammar</a>
<ul><li><a href="Re-Pair" title="Re-Pair">Re-Pair</a></li>
<li><a href="Sequitur_algorithm" title="Sequitur algorithm">Sequitur</a></li></ul></li>
<li><a href="Discrete_cosine_transform" title="Discrete cosine transform">LDCT</a></li>
<li><a href="Move-to-front_transform" title="Move-to-front transform">MTF</a></li>
<li><a href="PAQ" title="PAQ">PAQ</a></li>
<li><a href="Prediction_by_partial_matching" title="Prediction by partial matching">PPM</a></li>
<li><a href="Run-length_encoding" title="Run-length encoding">RLE</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Hybrid</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li>LZ77 + Huffman
<ul><li><a href="Deflate" title="Deflate">Deflate</a></li>
<li><a href="LZX" title="LZX">LZX</a></li>
<li><a href="Lempel%E2%80%93Ziv%E2%80%93Stac" title="Lempel–Ziv–Stac">LZS</a></li></ul></li>
<li>LZ77 + ANS
<ul><li><a href="LZFSE" title="LZFSE">LZFSE</a></li></ul></li>
<li>LZ77 + Huffman + ANS
<ul><li><a href="Zstd" title="Zstd">Zstandard</a></li></ul></li>
<li>LZ77 + Huffman + context
<ul><li><a href="Brotli" title="Brotli">Brotli</a></li></ul></li>
<li>LZSS + Huffman
<ul><li><a href="LHA_(file_format)" title="LHA (file format)">LHA/LZH</a></li></ul></li>
<li>LZ77 + Range
<ul><li><a href="LZMA" title="LZMA">LZMA</a></li>
<li>LZHAM</li></ul></li>
<li>RLE + BWT + MTF + Huffman
<ul><li><a href="Bzip2" title="Bzip2">bzip2</a></li></ul></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Lossy_compression" title="Lossy compression">Lossy</a><br>type</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Transform_coding" title="Transform coding">Transform</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Discrete_cosine_transform" title="Discrete cosine transform">Discrete cosine transform</a>
<ul><li><a href="Discrete_cosine_transform" title="Discrete cosine transform">DCT</a></li>
<li><a href="Modified_discrete_cosine_transform" title="Modified discrete cosine transform">MDCT</a></li></ul></li>
<li><a href="Discrete_sine_transform" title="Discrete sine transform">DST</a></li>
<li><a href="Fast_Fourier_transform" title="Fast Fourier transform">FFT</a></li>
<li><a href="Wavelet_transform" title="Wavelet transform">Wavelet</a>
<ul><li><a href="Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li>
<li><a href="Discrete_wavelet_transform" title="Discrete wavelet transform">DWT</a></li>
<li><a href="Set_partitioning_in_hierarchical_trees" title="Set partitioning in hierarchical trees">SPIHT</a></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Predictive</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li>
<ul><li><a href="Adaptive_differential_pulse-code_modulation" title="Adaptive differential pulse-code modulation">ADPCM</a></li></ul></li>
<li><a href="Linear_predictive_coding" title="Linear predictive coding">LPC</a>
<ul><li><a href="Algebraic_code-excited_linear_prediction" title="Algebraic code-excited linear prediction">ACELP</a></li>
<li><a href="Code-excited_linear_prediction" title="Code-excited linear prediction">CELP</a></li>
<li><a href="Log_area_ratio" title="Log area ratio">LAR</a></li>
<li><a href="Line_spectral_pairs" title="Line spectral pairs">LSP</a></li>
<li><a href="Warped_linear_predictive_coding" title="Warped linear predictive coding">WLPC</a></li></ul></li>
<li>Motion
<ul><li><a href="Motion_compensation" title="Motion compensation">Compensation</a></li>
<li><a href="Motion_estimation" title="Motion estimation">Estimation</a></li>
<li><a href="Motion_vector" class="mw-redirect" title="Motion vector">Vector</a></li></ul></li>
<li><a href="Psychoacoustics" title="Psychoacoustics">Psychoacoustic</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Data_compression#Audio" title="Data compression">Audio</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Bit_rate" title="Bit rate">Bit rate</a>
<ul><li><a href="Average_bitrate" title="Average bitrate">ABR</a></li>
<li><a href="Constant_bitrate" title="Constant bitrate">CBR</a></li>
<li><a href="Variable_bitrate" title="Variable bitrate">VBR</a></li></ul></li>
<li><a href="Companding" title="Companding">Companding</a></li>
<li><a href="Convolution" title="Convolution">Convolution</a></li>
<li><a href="Dynamic_range" title="Dynamic range">Dynamic range</a></li>
<li><a href="Latency_(audio)" title="Latency (audio)">Latency</a></li>
<li><a href="Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">Nyquist–Shannon theorem</a></li>
<li><a href="Sampling_(signal_processing)" title="Sampling (signal processing)">Sampling</a></li>
<li><a href="Silence_compression" title="Silence compression">Silence compression</a></li>
<li><a href="Sound_quality" title="Sound quality">Sound quality</a></li>
<li><a href="Speech_coding" title="Speech coding">Speech coding</a></li>
<li><a href="Sub-band_coding" title="Sub-band coding">Sub-band coding</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Audio_codec" title="Audio codec">Codec</a><br>parts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="A-law_algorithm" title="A-law algorithm">A-law</a></li>
<li><a href="%CE%9C-law_algorithm" title="Μ-law algorithm">μ-law</a></li>
<li>
<ul><li><a href="Adaptive_differential_pulse-code_modulation" title="Adaptive differential pulse-code modulation">ADPCM</a></li>
<li><a href="Delta_modulation" title="Delta modulation">DM</a></li></ul></li>
<li><a href="Fourier_transform" title="Fourier transform">FT</a>
<ul><li><a href="Fast_Fourier_transform" title="Fast Fourier transform">FFT</a></li></ul></li>
<li><a href="Linear_predictive_coding" title="Linear predictive coding">LPC</a>
<ul><li><a href="Algebraic_code-excited_linear_prediction" title="Algebraic code-excited linear prediction">ACELP</a></li>
<li><a href="Code-excited_linear_prediction" title="Code-excited linear prediction">CELP</a></li>
<li><a href="Log_area_ratio" title="Log area ratio">LAR</a></li>
<li><a href="Line_spectral_pairs" title="Line spectral pairs">LSP</a></li>
<li><a href="Warped_linear_predictive_coding" title="Warped linear predictive coding">WLPC</a></li></ul></li>
<li><a href="Modified_discrete_cosine_transform" title="Modified discrete cosine transform">MDCT</a></li>
<li><a href="Psychoacoustics" title="Psychoacoustics">Psychoacoustic model</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Image_compression" title="Image compression">Image</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Chroma_subsampling" title="Chroma subsampling">Chroma subsampling</a></li>
<li><a href="Coding_tree_unit" title="Coding tree unit">Coding tree unit</a></li>
<li><a href="Color_space" title="Color space">Color space</a></li>
<li><a href="Compression_artifact" title="Compression artifact">Compression artifact</a></li>
<li><a href="Image_resolution" title="Image resolution">Image resolution</a></li>
<li><a href="Macroblock" title="Macroblock">Macroblock</a></li>
<li><a href="Pixel" title="Pixel">Pixel</a></li>
<li><a href="Peak_signal-to-noise_ratio" title="Peak signal-to-noise ratio">PSNR</a></li>
<li><a href="Quantization_(image_processing)" title="Quantization (image processing)">Quantization</a></li>
<li><a href="Standard_test_image" title="Standard test image">Standard test image</a></li>
<li><a href="Texture_compression" title="Texture compression">Texture compression</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Methods</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Chain_code" title="Chain code">Chain code</a></li>
<li><a href="Discrete_cosine_transform" title="Discrete cosine transform">DCT</a></li>
<li><a href="Deflate" title="Deflate">Deflate</a></li>
<li><a href="Fractal_compression" title="Fractal compression">Fractal</a></li>
<li><a href="Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">KLT</a></li>
<li><a href="Pyramid_(image_processing)" title="Pyramid (image processing)">LP</a></li>
<li><a href="Run-length_encoding" title="Run-length encoding">RLE</a></li>
<li><a href="Wavelet_transform" title="Wavelet transform">Wavelet</a>
<ul><li><a href="Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li>
<li><a href="Discrete_wavelet_transform" title="Discrete wavelet transform">DWT</a></li>
<li><a href="Embedded_zerotrees_of_wavelet_transforms" title="Embedded zerotrees of wavelet transforms">EZW</a></li>
<li><a href="Set_partitioning_in_hierarchical_trees" title="Set partitioning in hierarchical trees">SPIHT</a></li></ul></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Data_compression#Video" title="Data compression">Video</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Bit_rate" title="Bit rate">Bit rate</a>
<ul><li><a href="Average_bitrate" title="Average bitrate">ABR</a></li>
<li><a href="Constant_bitrate" title="Constant bitrate">CBR</a></li>
<li><a href="Variable_bitrate" title="Variable bitrate">VBR</a></li></ul></li>
<li><a href="Display_resolution" title="Display resolution">Display resolution</a></li>
<li><a href="Film_frame" title="Film frame">Frame</a></li>
<li><a href="Frame_rate" title="Frame rate">Frame rate</a></li>
<li><a href="Video_compression_picture_types" title="Video compression picture types">Frame types</a></li>
<li><a href="Interlaced_video" title="Interlaced video">Interlace</a></li>
<li><a href="Video#Characteristics_of_video_streams" title="Video">Video characteristics</a></li>
<li><a href="Video_quality" title="Video quality">Video quality</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Video_codec" title="Video codec">Codec</a><br>parts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Discrete_cosine_transform" title="Discrete cosine transform">DCT</a></li>
<li><a href="Deblocking_filter" title="Deblocking filter">Deblocking filter</a></li>
<li><a href="Lapped_transform" title="Lapped transform">Lapped transform</a></li>
<li>Motion
<ul><li><a href="Motion_compensation" title="Motion compensation">Compensation</a></li>
<li><a href="Motion_estimation" title="Motion estimation">Estimation</a></li>
<li><a href="Motion_vector" class="mw-redirect" title="Motion vector">Vector</a></li></ul></li>
<li><a href="Wavelet_transform" title="Wavelet transform">Wavelet</a>
<ul><li><a href="Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li>
<li><a href="Discrete_wavelet_transform" title="Discrete wavelet transform">DWT</a></li></ul></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Information_theory" title="Information theory">Theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Compressed_data_structure" title="Compressed data structure">Compressed data structures</a>
<ul><li><a href="Compressed_suffix_array" title="Compressed suffix array">Compressed suffix array</a></li>
<li><a href="FM-index" title="FM-index">FM-index</a></li></ul></li>
<li><a href="Entropy_(information_theory)" title="Entropy (information theory)">Entropy</a></li>
<li><a href="Information_theory" title="Information theory">Information theory</a>
<ul><li><a href="Timeline_of_information_theory" title="Timeline of information theory">Timeline</a></li></ul></li>
<li><a href="Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li>
<li><a href="Prefix_code" title="Prefix code">Prefix code</a></li>
<li><a href="Quantization_(signal_processing)" title="Quantization (signal processing)">Quantization</a></li>
<li><a href="Rate%E2%80%93distortion_theory" title="Rate–distortion theory">Rate–distortion</a></li>
<li><a href="Redundancy_(information_theory)" title="Redundancy (information theory)">Redundancy</a></li>
<li><a href="Data_compression_symmetry" title="Data compression symmetry">Symmetry</a></li>
<li><a href="Smallest_grammar_problem" title="Smallest grammar problem">Smallest grammar problem</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Community</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Hutter_Prize" title="Hutter Prize">Hutter Prize</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Mark_Adler" title="Mark Adler">Mark Adler</a></li>
<li><a href="Phil_Katz" title="Phil Katz">Phil Katz</a></li></ul>
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