Bounds on the minimum distance of linear codes
Bounds on linear codes [84,4] over GF(2)
lower bound:  44 
upper bound:  44 
Construction
Construction of a linear code [84,4,44] over GF(2):
[1]: [16, 1, 16] Cyclic Linear Code over GF(2)
RepetitionCode of length 16
[2]: [32, 6, 16] Linear Code over GF(2)
PlotkinSum of [9] and [1]
[3]: [30, 4, 16] Linear Code over GF(2)
Shortening of [2] at { 31 .. 32 }
[4]: [44, 4, 23] Linear Code over GF(2)
Juxtaposition of [11] and [3]
[5]: [45, 4, 24] Linear Code over GF(2)
ExtendCode [4] by 1
[6]: [59, 4, 31] Linear Code over GF(2)
Juxtaposition of [11] and [5]
[7]: [60, 4, 32] Linear Code over GF(2)
ExtendCode [6] by 1
[8]: [8, 1, 8] Cyclic Linear Code over GF(2)
RepetitionCode of length 8
[9]: [16, 5, 8] Linear Code over GF(2)
PlotkinSum of [16] and [8]
[10]: [15, 5, 7] Linear Code over GF(2)
Puncturing of [9] at { 16 }
[11]: [14, 4, 7] Linear Code over GF(2)
Shortening of [10] at { 15 }
[12]: [74, 4, 39] Linear Code over GF(2)
Juxtaposition of [11] and [7]
[13]: [75, 4, 40] Linear Code over GF(2)
ExtendCode [12] by 1
[14]: [4, 1, 4] Cyclic Linear Code over GF(2)
RepetitionCode of length 4
[15]: [4, 3, 2] Cyclic Linear Code over GF(2)
Dual of the RepetitionCode of length 4
[16]: [8, 4, 4] Quasicyclic of degree 2 Linear Code over GF(2)
PlotkinSum of [15] and [14]
[17]: [7, 4, 3] Linear Code over GF(2)
Puncturing of [16] at 1
[18]: [82, 4, 43] Linear Code over GF(2)
Juxtaposition of [17] and [13]
[19]: [83, 4, 44] Linear Code over GF(2)
ExtendCode [18] by 1
[20]: [84, 4, 44] Linear Code over GF(2)
PadCode [19] by 1
last modified: 20010130
From Brouwer's table (as of 20070213)
Lb(84,4) = 44 is found by lengthening of:
Lb(83,4) = 44 is found by adding a parity check bit to:
Lb(82,4) = 43 is found by concatenation of
[7,4,3] and [75,4,40]codes
Ub(84,4) = 44 follows by the Griesmer bound.
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