\(q\)-ary duadic code[1][2][3][4]

Description

Member of a pair of cyclic linear binary codes that satisfy certain relations, depending on whether the pair is even-like or odd-like duadic. Duadic codes exist only when \(q\) is a square modulo \(n\) [2].

Duadic codes come in two pairs, an even-like duadic pair and an odd-like duadic pair. All codewords in the respective pairs are even-like, i.e., \(\sum_i c_i = 0\), or odd-like, i.e., \(\sum_i c_i \neq 0\). A code with all even-like (odd-like) codewords is called even-like (odd-like).

Duadic code pairs can be defined in terms of their idempotent generators (see Cyclic-to-polynomial correspondence). A pair of even-like codes \(C_1\) and \(C_2\) with respective idempotents \(e_1\) and \(e_2\) is an even-like duadic pair if (1) \(e_1(x)+e_2(x)=1-\frac{1}{n}(1+x+x^2+\cdots+x^{n-1})\) and (2) there exists a multiplier \(\mu\) such that \(C_1 \mu=C_2\) and \(C_2 \mu=C_1\).

There is an odd-like duadic pair \(\{D_1,D_2\}\) associated with the even-like pair \(\{C_1, C_2\}\), where \(1-e_2(x)\) generates \(D_1\) and \(1-e_1(x)\) generates \(D_2\). The even-pair codes are \([n,\frac{n-1}{2}]_q\) codes while the odd-pair codes are \([n,\frac{n+1}{2}]_q\) codes.

Notes

Reviews of duadic codes [2][5].

Parent

Child

Cousin

References

[1]
V. Pless, “Q-codes”, Journal of Combinatorial Theory, Series A 43, 258 (1986). DOI
[2]
V. Pless, “Duadic Codes and Generalizations”, Eurocode ’92 3 (1993). DOI
[3]
J. J. Rushanan, Topics in Integral Matrices and Abelian Group Codes, California Institute of Technology, 1986. DOI
[4]
M. Smid, “Duadic codes (Corresp.)”, IEEE Transactions on Information Theory 33, 432 (1987). DOI
[5]
W. C. Huffman and V. Pless, Fundamentals of Error-correcting Codes (Cambridge University Press, 2003). DOI

Zoo code information

Internal code ID: q-ary_duadic

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Zoo Code ID: q-ary_duadic

Cite as:
“\(q\)-ary duadic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_duadic
BibTeX:
@incollection{eczoo_q-ary_duadic, title={\(q\)-ary duadic code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/q-ary_duadic} }
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Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/q-ary_digits/cyclic/q-ary_duadic.yml.