Also known as Pless double circulant code.

## Description

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.

## Rate

Achieve capacity of the binary erasure channel; see Ref. [5].

## Parent

## Cousins

- 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 [2,6,7].

## References

- [1]
- V. Pless, On a new family of symmetry codes and related new five-designs, BAMS 75 (1969), 1339-1342
- [2]
- V. Pless, “Symmetry codes over GF(3) and new five-designs”, Journal of Combinatorial Theory, Series A 12, 119 (1972) DOI
- [3]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [4]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [5]
- K. Ivanov and R. L. Urbanke, “Capacity-achieving codes: a review on double transitivity”, (2020) arXiv:2010.15453
- [6]
- 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
- [7]
- M. HUBER, “CODING THEORY AND ALGEBRAIC COMBINATORICS”, Selected Topics in Information and Coding Theory 121 (2010) arXiv:0811.1254 DOI

## Page edit log

- Connor Clayton (2024-03-15) — most recent
- Victor V. Albert (2022-07-14)

## Cite as:

“Pless symmetry code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/pless_symmetry