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Indata compression, auniversal code for integers is aprefix code that maps thepositive integers onto binary codewords, with the additional property that whatever the trueprobability distribution on integers, as long as the distribution is monotonic (i.e.,p(i) ≥ p(i + 1) for all positive i), theexpected lengths of the codewords are within a constant factor of the expected lengths that theoptimal code for that probability distribution would have assigned. A universal code isasymptotically optimal if the ratio between actual and optimalexpected lengths is bounded by a function of theinformation entropy of the code that, in addition to being bounded, approaches 1 as entropy approaches infinity.

In general, most prefix codes for integers assign longer codewords to larger integers. Such a code can be used to efficiently communicate a message drawn from a set of possible messages, by simply ordering the set of messages by decreasing probability and then sending the index of the intended message. Universal codes are generally not used for precisely known probability distributions, and no universal code is known to be optimal for any distribution used in practice.

A universal code should not be confused withuniversal source coding, in which the data compression method need not be a fixed prefix code and the ratio between actual and optimal expected lengths must approach one. However, note that an asymptotically optimal universal code can be used onindependent identically-distributed sources, by using increasingly largeblocks, as a method of universal source coding.

These are some universal codes for integers; an asterisk (*) indicates a code that can be trivially restated inlexicographical order, while a double dagger (‡) indicates a code that is asymptotically optimal:

• Elias gamma coding *
• Elias delta coding * ‡
• Elias omega coding *^{[further explanation needed]} ‡
• Exp-Golomb coding *, which has Elias gamma coding as a special case. (Used inH.264/MPEG-4 AVC)
• Fibonacci coding
• Levenshtein coding * ‡, the original universal coding technique[1]
• Byte coding where a special bit pattern (with at least two bits) is used to mark the end of the code — for example, if an integer is encoded as a sequence ofnibbles representing digits inbase 15 instead of the more naturalbase 16, then the highest nibble value (i.e., a sequence of four ones in binary) can be used to indicate the end of the integer.
• Variable-length quantity

These are non-universal ones:

• Unary coding, which is used in Elias codes
• Rice coding, which is used in theFLACaudio codec and which has unary coding as a special case
• Golomb coding, which has Rice coding and unary coding as special cases.

Their nonuniversality can be observed by noticing that, if any of these are used to code theGauss–Kuzmin distribution or theZeta distribution with parameter s=2, expected codeword length is infinite. For example, using unary coding on the Zeta distribution yields an expected length of

${\displaystyle E(l)=\frac{6}{{\pi }^2}\sum _{l=1}^{\mathrm{\infty }}\frac{1}{l}=\mathrm{\infty }.}$

On the other hand, using the universal Elias gamma coding for the Gauss–Kuzmin distribution results in an expected codeword length (about 3.51 bits) near entropy (about 3.43 bits)- Академия Google.

Huffman coding andarithmetic coding (when they can be used) give at least as good, and often better compression than any universal code.

However, universal codes are useful when Huffman coding cannot be used — for example, when one does not know the exact probability of each message, but only knows the rankings of their probabilities.

Universal codes are also useful when Huffman codes are inconvenient. For example, when the transmitter but not the receiver knows the probabilities of the messages, Huffman coding requires an overhead of transmitting those probabilities to the receiver. Using a universal code does not have that overhead.

Each universal code, like each other self-delimiting (prefix) binary code, has its own "implied probability distribution" given byP(i)=2^−l(i) wherel(i) is the length of theith codeword andP(i) is the corresponding symbol's probability. If the actual message probabilities areQ(i) andKullback–Leibler divergence${\displaystyle D_{\text{KL}}(Q\Vert P)}$ is minimized by the code withl(i), then the optimal Huffman code for that set of messages will be equivalent to that code. Likewise, how close a code is to optimal can be measured by this divergence. Since universal codes are simpler and faster to encode and decode than Huffman codes (which is, in turn, simpler and faster thanarithmetic encoding), the universal code would be preferable in cases where${\displaystyle D_{\text{KL}}(Q\Vert P)}$ is sufficiently small.Lossless Data Compression Program: Hybrid LZ77 RLE

For anygeometric distribution (an exponential distribution on integers), a Golomb code is optimal. With universal codes, the implicit distribution is approximately apower law such as${\displaystyle 1/n^2}$ (more precisely, aZipf distribution).For theFibonacci code, the implicit distribution is approximately${\displaystyle 1/n^q}$, with

${\displaystyle q=1/{\log }_2(\phi )\simeq 1.44,}$

where${\displaystyle \phi }$ is thegolden ratio. For the ternarycomma code (i.e., encoding in base 3, represented with 2 bits per symbol), the implicit distribution is a power law with${\displaystyle q=1+{\log }_3(4/3)\simeq 1.26}$. These distributions thus have near-optimal codes with their respective power laws.

• Data Compression, by Debra A. Lelewer and Daniel S. Hirschberg (University of California, Irvine)
• Information Theory, Inference, and Learning Algorithms, byDavid MacKay, has a chapter on codes for integers, including an introduction to Elias codes.
• Кодирование целых чисел has mostly English-language papers on universal and other integer codes.

• Data compression
• Entropy coding

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