Bounds on the minimum distance of linear codes
Bounds on linear codes [52,9] over GF(4)
Construction of a linear code [52,9,32] over GF(4):
: [51, 9, 31] Cyclic Linear Code over GF(2^2)
CyclicCode of length 51 with generating polynomial x^42 + w*x^41 + x^40 + x^38 + x^37 + x^36 + w*x^34 + w*x^32 + x^31 + w*x^29 + w^2*x^28 + x^25 + w^2*x^24
+ w^2*x^23 + w*x^22 + w*x^19 + w*x^17 + w*x^16 + w*x^15 + w*x^14 + w*x^13 + w*x^12 + w^2*x^11 + w*x^10 + w^2*x^9 + w^2*x^7 + w^2*x^6 + x^3 + w*x^2 + w*x + 1
: [52, 9, 32] Linear Code over GF(2^2)
Append to the code  the column vector ( w 0 w^2 w^2 w^2 1 w 1 1)
last modified: 2002-03-28
From Brouwer's table (as of 2007-02-13)
Lb(52,9) = 32 DaH
Ub(52,9) = 33 follows by a one-step Griesmer bound from:
Ub(18,8) = 8 follows by a one-step Griesmer bound from:
Ub(9,7) = 2 is found by considering shortening to:
Ub(6,4) = 2 is found by construction B:
[consider deleting the (at most) 4 coordinates of a word in the dual]
Rumen Daskalov & Plamen Hristov, New One-Generator Quasi-Cyclic Codes over
GF(7), preprint, Oct 2001. R. Daskalov & P Hristov, New One-Generator
Quasi-Twisted Codes over GF(5), (preprint) Oct. 2001. R. Daskalov & P Hristov,
New Quasi-Twisted Degenerate Ternary Linear Codes, preprint, Nov 2001.
- 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