Pless symmetry code[1,2] 

Also known as Pless double circulant code.


A member of a family of self-dual ternary \([2q+2,q+1]_3\) codes for any power of an odd prime satisfying \(q \equiv 2\) modulo 3.

The code's generator matrix is \(G = [I | S_q]\), where \(I\) is the \((q+1)\)-dimensional identity matrix, and where the matrix \(S_q\) is shown in the following image (with \(q=p\)). There, \(\chi(0)=0\), \(\chi(x)=1\) if \(x\) is a square in \(GF(q)\) and, \(\chi(x)=-1\) if \(x\) is a not square in \(GF(q)\).

See [3; Sec. 10.5][4; pg. 87] for more details.



  • Ternary Golay code — The Pless symmetry code for \(p=5\) is the extended ternary Golay code.
  • Combinatorial design — The supports of fixed-weight codewords of certain Pless symmetry codes support combinatorial designs [2,5,6].


V. Pless, On a new family of symmetry codes and related new five-designs, BAMS 75 (1969), 1339-1342
V. Pless, “Symmetry codes over GF(3) and new five-designs”, Journal of Combinatorial Theory, Series A 12, 119 (1972) DOI
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
V. Pless, “The Weight of the Symmetry Code for p=29 and the 5‐Designs Contained Therein”, Annals of the New York Academy of Sciences 175, 310 (1970) DOI
M. HUBER, “CODING THEORY AND ALGEBRAIC COMBINATORICS”, Selected Topics in Information and Coding Theory 121 (2010) arXiv:0811.1254 DOI
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Zoo Code ID: pless_symmetry

Cite as:
“Pless symmetry code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.
@incollection{eczoo_pless_symmetry, title={Pless symmetry code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Pless symmetry code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.