Bounds on the minimum distance of linear codes
Bounds on linear codes [249,21] over GF(2)
lower bound:  106 
upper bound:  112 
Construction
Construction type: GraX
Construction of a linear code [249,21,106] over GF(2):
[1]: [9, 8, 2] Cyclic Linear Code over GF(2)
Dual of the RepetitionCode of length 9
[2]: [240, 13, 112] "Goppa code (r = 102)" Linear Code over GF(2)
GoppaCode with 240 points over GF(256) and polynomial x^102 + x^72 + x^42 + x^12 where w := GF(256).1
[3]: [240, 21, 104] "Goppa code (r = 96)" Linear Code over GF(2)
GoppaCode with 240 points over GF(256) and polynomial x^96 + x^66 + x^36 + x^6 where w := GF(256).1
[4]: [249, 21, 106] Linear Code over GF(2)
ConstructionX using [3] [2] and [1]
last modified: 20201018
From Brouwer's table (as of 20070213)
Lb(249,21) = 105 is found by truncation of:
Lb(256,21) = 112 XBC
Ub(249,21) = 113 follows by a onestep Griesmer bound from:
Ub(135,20) = 56 is found by considering shortening to:
Ub(132,17) = 56 otherwise adding a parity check bit would contradict:
Ub(133,17) = 57 LP
References
LP:
Follows from the linear programming bound.
XBC:
Extended BCH code.
Notes
 All codes establishing the lower bounds were constructed using
MAGMA.
 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 20012006.
 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 email:
codes [at] codetables.de

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Last change: 30.12.2011