Bounds on the minimum distance of linear codes
Bounds on linear codes [56,9] over GF(2)
ConstructionConstruction type: Chen
Construction of a linear code [56,9,24] over
: [56, 9, 24] Quasicyclic of degree 4 Linear Code over GF(2)
QuasiCyclicCode of length 56 with Generating Polynomials (in octal):
[ 4465, 2307, 2133, 3145 ]
last modified: 2001-01-30
From Brouwer's table (as of 2007-02-13)
Lb(56,9) = 24 is found by taking a subcode of:
Lb(56,10) = 24 BZ
Ub(56,9) = 24 follows by a one-step Griesmer bound from:
Ub(31,8) = 12 is found by considering shortening to:
Ub(30,7) = 12 otherwise adding a parity check bit would contradict:
Ub(31,7) = 13 vT3
E. L. Blokh & V. V. Zyablov, Coding of generalized concatenated codes,
Probl. Inform. Transm. 10 (1974) 218-222.
H.C.A. van Tilborg, The smallest length of binary 7-dimensional linear codes
with prescribed minimum distance, Discr. Math. 33 (1981) 197-207.
- All codes establishing the lower bounds were constructed using
- Upper bounds are taken from the tables of Andries E. Brouwer, with the exception of codes over GF(7) with n>50.
For most of these codes, the upper bounds are rather weak.
Upper bounds for codes over GF(7) with small dimension have been provided by Rumen Daskalov.
- Special thanks to John Cannon for his support in this project.
- A prototype version of MAGMA's code database over GF(2) was
written by Tat Chan in 1999 and extended later that year by
Damien Fisher. The current release version was
developed by Greg White over the period 2001-2006.
- Thanks also to Allan Steel for his MAGMA support.
- My apologies to all authors that have contributed codes to this table for not giving specific credits.
- If you have found any code improving the bounds or some errors, please send me an e-mail:
codes [at] codetables.de
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Last change: 30.12.2011