Karlin double circulant code

Karlin double circulant code[1]

Description

Member of the family of \([2m+2,m+1]\) double circulant codes such that \(m\) is prime of the form \(8k+3\) for some \(k\), and \(2m+2\) is a multiple of eight. See [2; Ch. 16] for their generator matrix. Karlin double circulant codes can be mapped to extended cyclic and extended quadratic-residue codes over \(GF(4)\) [3,4][2; Ch. 16][5; Sec. 2.4.2] by identifying \((0,\omega,\bar{\omega},1)\) with \((00),(10),(01),(11)\) [3].

Cousin

Quadratic-residue (QR) code— Karlin double circulant codes can be mapped to extended cyclic and extended quadratic-residue codes over \(GF(4)\) [3,4][2; Ch. 16][5; Sec. 2.4.2] by identifying \((0,\omega,\bar{\omega},1)\) with \((00),(10),(01),(11)\) [3].

Member of code lists

Primary Hierarchy

Classical Domain

Binary Kingdom

Binary codeECC

Binary group-orbit codeECC

Linear binary codeLinear \(q\)-ary Linear code over \(\mathbb{Z}_q\) ECC

Parents

Linear binary code

Quasi-cyclic codeECC

Karlin double circulant codes can be mapped to extended cyclic and extended quadratic-residue codes over \(GF(4)\) [3,4][2; Ch. 16][5; Sec. 2.4.2] by identifying \((0,\omega,\bar{\omega},1)\) with \((00),(10),(01),(11)\) [3].

Self-dual linear codeSelf-dual additive Linear \(q\)-ary ECC

Karlin double circulant codes are Euclidean self-dual doubly even codes [2; Ch. 16], and some of them are extremal [6,7].

Karlin double circulant code

Children

\([24, 12, 8]\) Extended Golay code

The extended Golay code is equivalent to the Karlin double circulant code for \(m=11\) [2; Ch. 16].

\([2^r,2^r-r-1,4]\) Extended Hamming code

The extended Hamming code is equivalent to the Karlin double circulant code for \(m=3\) [2; Ch. 16].

References

[1]: M. Karlin, “New binary coding results by circulants”, IEEE Transactions on Information Theory 15, 81 (1969) DOI
[2]: F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
[3]: P. Gaborit, V. Pless, P. Solé, and O. Atkin, “Type II Codes over F4”, Finite Fields and Their Applications 8, 171 (2002) DOI
[4]: Karlin M, MacWilliams FJ. Quadratic residue codes over GF (4) and their binary images. InIEEE Int. Symp. on Information Theory, Asilomar, CA 1972.
[5]: Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
[6]: M. Karlin, V. K. Bhargava, and S. E. Tavares, “A note on extended quaternary quadratic residue codes and their binary images”, Information and Control 38, 148 (1978) DOI
[7]: K. Betsumiya, T. A. Gulliver, M. Harada, and A. Munemasa, “On Type II codes over F/sub 4/”, IEEE Transactions on Information Theory 47, 2242 (2001) DOI

Page edit log

Victor V. Albert (2025-01-08) — most recent

Cite as:

“Karlin double circulant code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/karlin

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/cyclic/karlin.yml.