## Bounds on the minimum distance of linear codes

### Bounds on linear codes [40,9] over GF(2)

 lower bound: 16 upper bound: 16

### Construction

```Construction of a linear code [40,9,16] over GF(2):
[1]:  [4, 1, 4] Cyclic Linear Code over GF(2)
RepetitionCode of length 4
[2]:  [4, 3, 2] Cyclic Linear Code over GF(2)
Dual of the RepetitionCode of length 4
[3]:  [8, 4, 4] Quasicyclic of degree 2 Linear Code over GF(2)
PlotkinSum of [2] and [1]
[4]:  [39, 10, 15] Linear Code over GF(2)
Let C1 be the BCHCode over GF( 2) of parameters 31 11. Let C2 the SubcodeBetweenCode of
dimension 10 between C1 and the BCHCode with
parameters 31 15. Return ConstructionX using C1, C2 and [3]
[5]:  [40, 10, 16] Linear Code over GF(2)
ExtendCode [4] by 1
[6]:  [40, 9, 16] Linear Code over GF(2)
Subcode of [5]

last modified: 2001-01-30
```

### From Brouwer's table (as of 2007-02-13)

```Lb(40,9) = 16 is found by taking a subcode of:
Lb(40,10) = 16 is found by adding a parity check bit to:
Lb(39,10) = 15 X

Ub(40,9) = 16 follows by a one-step Griesmer bound from:
Ub(23,8) = 8 follows by a one-step Griesmer bound from:
Ub(14,7) = 4 follows by a one-step Griesmer bound from:
Ub(9,6) = 2 is found by considering shortening to:
Ub(8,5) = 2 is found by construction B:
[consider deleting the (at most) 4 coordinates of a word in the dual]
```
X:

### 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 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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This page is maintained by Markus Grassl (grassl@ira.uka.de). Last change: 30.12.2011