## Description

## Protection

The standard metric for Galois-field \(q\)-ary codes is the Hamming metric, but other metrics also exist [1]. A code 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.

Weight enumerator and four fundamental parameters: Determining protection and bounds on code parameters can also be done using the code's weight enumerator (cf. quantum weight enumerators), \begin{align} \begin{split} A(x,y)&=\sum_{j=0}^{n}A_{j}x^{n-j}y^{j}\\ A_{j}&=\text{number of wt-}j\text{ codewords}~. \end{split} \tag*{(1)}\end{align} The weight enumerator and it's Fourier transform the dual weight enumerator satisfy the MacWilliams identity [2,3]; see [4; Ch. 5].

The distance of the code is the value of the first nonzero coefficient \(A_j\), while the dual distance is the first nonzero coefficient of the dual weight enumerator. The number of the code is the number of nonzero \(A_j\)'s, corresponding to the number of distinct nonzero distances between codewords. The external distance is the number of nonzero coefficients of the dual weight enumerator. The distance, dual distance, number and external distance make up the four fundamental parameters of a code [5][4; Ch. 5].

Other types of weight enumerators includes the Hamming weight enumerator, Lee weight enumerator, joint weight enumerator, split weight enumerator, and biweight enumerator [4].

## Rate

## Decoding

## Notes

## Parent

- Ring code — Galois fields are rings under addition.

## Children

## Cousins

- Combinatorial design — Designs can be constructed from \(q\)-ary codes by taking the supports of a subset of codewords of constant weight.
- Traceability code — A \(q\)-ary code with distance \(d \geq n(1-1/t^2)\) has the \(t\)-traceability property [7; Thm. 4.3].
- Convolutional code — Convolutional codes for finite block size are \(q\)-ary codes.
- Polyphase code — Polyphase codes are spherical codes that can be obtained from \(q\)-ary codes.

## References

- [1]
- M. Grassl, A.-L. Horlemann, and V. Weger, “The Subfield Metric and its Application to Quantum Error Correction”, Journal of Algebra and Its Applications (2023) arXiv:2212.00431 DOI
- [2]
- J. Macwilliams, “A Theorem on the Distribution of Weights in a Systematic Code†”, Bell System Technical Journal 42, 79 (1963) DOI
- [3]
- F. J. MacWilliams, N. J. A. Sloane, and J.-M. Goethals, “The MacWilliams Identities for Nonlinear Codes”, Bell System Technical Journal 51, 803 (1972) DOI
- [4]
- F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
- [5]
- P. Delsarte, “Four fundamental parameters of a code and their combinatorial significance”, Information and Control 23, 407 (1973) DOI
- [6]
- 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.
- [7]
- J. N. Staddon, D. R. Stinson, and Ruizhong Wei, “Combinatorial properties of frameproof and traceability codes”, IEEE Transactions on Information Theory 47, 1042 (2001) DOI

## Page edit log

- Victor V. Albert (2022-02-16) — most recent
- Victor V. Albert (2021-10-29)

## Cite as:

“Galois-field \(q\)-ary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_digits_into_q-ary_digits