Cyclic linear binary code


A binary code of length \(n\) is cyclic if, for each codeword \(c_1 c_2 \cdots c_n\), the cyclically shifted string \(c_n c_1 \cdots c_{n-1}\) is also a codeword. A cyclic code is called primitive when \(n=2^r-1\) for some \(r\geq 2\). A shortened cyclic code is obtained from a cyclic code by taking only codewords with the first \(j\) zero entries, and deleting those zeroes.


Shift bound [1] gives a lower bound on the distance of cyclic binary codes.


Meggitt decoder [2].


See Ch. 7 of Ref. [3] for an exposition on cyclic codes.




  • Majorana stabilizer code — Cyclic binary linear codes can be used to construct translation-invariant Majorana stabilizer codes, provided that they are also self-orthogonal [4].
  • Reed-Muller (RM) code — Punctured RM codes are cyclic ([3], Ch. 13, Thm. 11), making RM codes extended cyclic codes. RM codes with nonzero evaluation points are cyclic ([5], pg. 52).


J. van Lint and R. Wilson, “On the minimum distance of cyclic codes”, IEEE Transactions on Information Theory 32, 23 (1986). DOI
J. Meggitt, “Error correcting codes and their implementation for data transmission systems”, IEEE Transactions on Information Theory 7, 234 (1961). DOI
F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
Sagar Vijay and Liang Fu, “Quantum Error Correction for Complex and Majorana Fermion Qubits”. 1703.00459
M. A. Tsfasman and S. G. Vlăduţ, Algebraic-geometric Codes (Springer Netherlands, 1991). DOI

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“Cyclic linear binary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_binary_cyclic, title={Cyclic linear binary code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Cyclic linear binary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.