## Bounds on the minimum distance of linear codes

### Bounds on linear codes [232,13] over GF(2)

 lower bound: 106 upper bound: 109

### Construction

```Construction of a linear code [232,13,106] over
GF(2):
:  [4, 4, 1] Cyclic Linear Code over GF(2)
UniverseCode of length 4
:  [15, 2, 14] Linear Code over GF(2^4)
Shortening of  at { 16 .. 17 }
:  [225, 9, 105] Linear Code over GF(2)
ZinovievCode using inner codes:  , outer codes:  
:  [15, 1, 15] Cyclic Linear Code over GF(2)
RepetitionCode of length 15
:  [17, 4, 14] "BCH code (d = 14, b = 11)" Linear Code over GF(2^4)
BCHCode over GF(16) with parameters 17 14 11
:  [16, 3, 14] Linear Code over GF(2^4)
Shortening of  at { 17 }
:  [15, 3, 13] Linear Code over GF(2^4)
Puncturing of  at { 16 }
:  [15, 5, 7] "BCH code (d = 7, b = 1)" Linear Code over GF(2)
BCHCode with parameters 15 7
:  [15, 4, 8] "BCH code (d = 8, b = 15)" Linear Code over GF(2)
BCHCode with parameters 15 8 0
: [225, 13, 104] Linear Code over GF(2)
ZinovievCode using inner codes:  , outer codes:  
: [229, 13, 105] Linear Code over GF(2)
ConstructionX using   and 
: [230, 13, 106] Linear Code over GF(2)
ExtendCode  by 1
: [232, 13, 106] Linear Code over GF(2)

```

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

```Lb(232,13) = 106 is found by truncation of:
Lb(234,13) = 108 is found by adding a parity check bit to:
Lb(233,13) = 107 Gra

Ub(232,13) = 110 follows by a one-step Griesmer bound from:
Ub(121,12) = 55 Bro
```
###### References
Bro: A.E. Brouwer, The linear programming bound for binary linear codes, IEEE Trans. Inform. Th. 39 (1993) 677-680.

Gra:

### 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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