Galois-field \(q\)-ary code


Encodes \(K\) states (codewords) in \(n\) \(q\)-ary coordinates over the field \(GF(q)=\mathbb{F}_q\) and has distance \(d\). Usually denoted as \((n,K,d)_q\). The distance is the minimum number of coordinates where two strings in the code differ.


Detects errors on up to \(d-1\) coordinates, corrects erasure errors on up to \(d-1\) coordinates, and corrects general errors on up to \(\left\lfloor (d-1)/2 \right\rfloor\) coordinates.


For small \(n\), decoding can be based on a lookup table. For infinite code families, the size of such a table scales exponentially with \(n\), so approximate decoding algorithms scaling polynomially with \(n\) have to be used. The decoder determining the most likely error given a noise channel is called the maximum-likelihood decoder.Given a received string \(x\) and an error bound \(e\), a list decoder returns a list of all codewords that are at most \(e\) from \(x\). The number of codewords in a neighborhood of \(x\) has to be polynomial in \(n\) in order for this decoder to run in time polynomial in \(n\).


Tables of bounds and examples of linear codes for various \(n\) and \(k\), extending code tables by Brouwer [1], are maintained by M. Grassl at this website.




  • Group-based code — Group-based codes whose alphabet is based on the field \(GF(q)\), taken to be an abelian group under addition, are \(q\)-ary codes.
  • Polyphase code — Polyphase codes are spherical codes that can be obtained from \(q\)-ary codes.


Andries E. Brouwer, Bounds on linear codes, in: Vera S. Pless and W. Cary Huffman (Eds.), Handbook of Coding Theory, pp. 295-461, Elsevier, 1998.
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Cite as:
“Galois-field \(q\)-ary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_q-ary_digits_into_q-ary_digits, title={Galois-field \(q\)-ary code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Galois-field \(q\)-ary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.