## Bounds on the minimum distance of linear codes

### Bounds on linear codes [136,9] over GF(4)

 lower bound: 89 upper bound: 96

### Construction

```Construction of a linear code [136,9,89] over GF(4):
[1]:  [137, 9, 90] Linear Code over GF(2^2)
Code found by Axel Kohnert
Construction from a stored generator matrix:

[ 1, 0, 0, 0, 0, 0, 0, 0, 0, w^2, w^2, w^2, w^2, 0, 0, w, w^2, 0, 1, w^2, 0, 1, 1, w^2, w, 0, 1, w, w^2, w^2, 1, w^2, 1, 0, w^2, 0, 0, 1, 1, w, 0, 0, w, 0, w,
1, 1, w^2, w, 0, w^2, 1, w^2, w, 1, w, 1, w^2, w, w, w^2, w^2, 0, w, 0, 1, 1, 0, w, w, 0, 0, w, w^2, w^2, 0, 0, w^2, w^2, 0, w^2, 0, 1, 0, 0, 0, w^2, w^2, w, 0,
0, 0, 1, 0, w, 1, w^2, 0, w^2, w, 1, w^2, w, w^2, 0, 1, w, w, 1, 1, 0, w, w, 0, 0, 1, w, w^2, 1, w, w^2, 1, 0, w, w^2, 0, w, 1, w, w, w^2, w, 1, 0, w^2, 0, 1 ]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0, w^2, w, w^2, w^2, 1, w, w^2, w, 1, w, 1, w^2, 0, w, 1, w^2, 1, w, w^2, w^2, 1, w^2, 1, w^2, 0, 0, 0, w, 1, 0, w^2, w^2, w^2, 0, 0,
1, 1, w^2, 0, 0, 0, w^2, w, w^2, 1, w, w^2, w, w, 0, 0, w, w, w, w^2, w^2, w^2, 0, 1, w^2, w, 1, w, 0, 0, 1, 1, 1, 0, 1, w^2, 0, w^2, 0, w^2, 0, w, 0, w, 1,
w^2, 0, w, 1, w, 0, 0, 1, 1, 0, 1, 0, w, w, 0, w, w, w, w^2, 0, 1, 1, w^2, w^2, 1, w^2, 0, 1, 0, 1, 1, w, 1, 1, 0, w^2, w^2, 1, 0, w^2, w^2, 0, 0, 1, 0, w^2, 0,
w^2 ]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0, w^2, 0, 0, 1, w^2, 1, w^2, 1, w^2, w^2, w^2, w^2, 1, w^2, 1, 1, 0, 1, w^2, 1, w^2, 0, w, 1, w, 0, w, 1, w^2, 1, w^2, w^2, w^2, 0,
0, w, 1, w, w, w^2, 1, 0, 0, w, 1, 0, 1, 1, w, 0, w, 0, 1, 1, w, w, 0, w^2, w, 1, w^2, w^2, 0, w^2, w^2, w^2, w^2, w, w, w, 0, w, w, w^2, w^2, 1, 0, w^2, w^2,
w^2, 1, 0, w, 0, w^2, w, 1, w, 0, w^2, 1, 1, w^2, 1, w^2, 1, 0, w, 0, 0, w^2, w, 1, 1, w, 1, 1, 1, w^2, 0, 1, 1, 1, 1, 1, w, 1, 1, w^2, 1, 0, 1, 1, 1, 1, 0, w,
w^2 ]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, w^2, 1, 0, 1, w^2, w^2, 1, 1, w^2, 1, 0, 0, 1, 0, w^2, 1, w^2, 1, w, w^2, w, w^2, 0, w, 1, 1, 1, w, 1, w^2, 0, w^2, w^2, w^2,
w^2, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, w, w, 0, w, w^2, 0, 0, 1, w, w, 0, 0, w, w, w, 0, 0, 1, w, w^2, 1, w, 1, 0, w, 0, 0, 0, 0, 0, w^2, 0, w^2, 0, w, w, 0, w, w,
0, w, 0, w^2, 0, w^2, w^2, w, 0, 1, w^2, w^2, 1, 1, 1, 0, w, w, 1, 1, w^2, 1, 1, w^2, w, 0, 0, w^2, w, w^2, w^2, 1, w, 1, 1, w, 1, 1, 1, 0, w^2, 0, w^2 ]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, w^2, 0, 1, w, w, w, 0, w, w^2, w^2, w, 0, 0, w, w^2, w^2, 1, w, w^2, 1, 0, 1, 1, w^2, w, w, w^2, 0, 0, w^2, 1, 1, 1, w^2,
w, 1, w, 1, 1, 1, w^2, w^2, 0, w, 1, 1, w^2, w^2, 0, 0, 1, w, 0, 0, 1, w, w, 1, 0, 1, 1, w^2, 1, w^2, 0, 1, 1, 0, 1, w, w^2, 1, 0, w^2, 0, 0, w, w, 0, w, 0,
w^2, 0, 1, w, w, 1, 0, 1, w, w^2, w^2, 0, w^2, 0, w^2, w, 0, w, 0, 1, 1, 1, w, w^2, 0, 0, w^2, w^2, 1, 1, 1, 1, 1, w, w, w, 0, 0, 1, w, w^2, w, w, w^2 ]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0, w, w^2, w^2, w^2, w, 1, 1, 1, w^2, 0, 0, 0, w, w, w^2, w^2, 0, w^2, w^2, 1, 1, 1, w, w^2, w^2, 1, w^2, 0, 0, 0, 1, w, 1, w, w, 0,
w, w, 0, 0, w^2, w, 1, 1, 1, 0, w^2, 0, 0, w, w, 0, w, 0, 1, w^2, w^2, 1, w, 1, w^2, 1, 0, w, w^2, 0, 0, w, w, w, 1, 1, 1, 0, 0, 0, w^2, 1, w^2, 1, w, w, w^2,
1, w, 0, w, 1, w, w^2, 0, w, w^2, 0, w, 0, w, w, 1, w, 1, w^2, 0, 1, 0, w^2, 0, 1, 0, w^2, w^2, 0, 0, w, 1, w, w, w, 0, w^2, w, w^2, w^2, 1, 0, w^2, w^2, w^2 ]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, w, 1, w, 1, 1, w, w, 0, 1, w, 1, w, 1, w, w^2, 0, 0, 1, 0, 1, w, 1, 1, 0, w^2, 0, 0, 0, 1, w^2, w^2, w^2, w, w^2, 1, w, 0, w^2,
w, 0, 1, w, 1, w^2, 1, w, 1, 0, w, 1, w^2, 0, w^2, w^2, w, 0, w, 1, w^2, 0, w, w^2, w^2, 0, 1, w^2, 1, w, 0, w, w^2, 0, 0, w^2, w^2, 1, w^2, w^2, 1, w, w^2, 1,
0, w, 0, w, 0, 0, 1, w, 0, 1, 1, w, 0, w^2, 0, 1, w^2, w^2, w^2, 1, w, 0, w^2, 1, 0, 0, w^2, 0, w, 0, 0, w, w^2, 0, 0, w, w, w^2, 1, w^2, 0, 1, 1, 0, 0 ]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, w, w, 1, 0, w, 0, w, 1, 1, 1, w, 1, 1, 1, 0, w^2, w, 0, w, 0, w^2, w^2, 1, 0, w^2, w, w^2, 0, w^2, w, w^2, 0, 0, w^2, 0, 1, w, w^2,
w, 0, 0, 1, w^2, 0, w^2, 0, w, 1, 1, 0, w, w, 1, w, 0, w, 0, w^2, w, w, 0, w^2, w, 0, 0, w^2, 0, 0, w, 1, 0, 0, 1, 0, w, w, w^2, 1, 1, w, 0, 0, 1, 1, 1, 1, w,
w, w^2, w, w, w^2, 1, w^2, w^2, w^2, 0, 0, 0, 1, w, w^2, 1, 0, w, 1, 0, 0, w, 1, 0, w, 1, w, w^2, w, 0, w^2, 1, 1, w, 1, 0, 0, 1, w^2, w, 1 ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1, w^2, 0, w^2, w, w, w, w, 1, w, 0, 0, 0, w, w^2, 1, 0, 0, w^2, 0, w, 0, 0, w^2, w, 0, 0, 0, w, 0, 1, 0, w^2, 0, 0, w, 1, 0, 1, w^2,
w, w^2, w, 1, 0, w, 1, w, w^2, w^2, 0, w, 0, w^2, 1, 1, w, w, w^2, w, 1, w^2, w^2, 0, w, 1, w, w^2, w, 1, 1, 0, 0, 1, 1, w, 1, 1, 0, 1, w^2, 1, w^2, w^2, w,
w^2, w^2, 1, w, 1, w^2, w^2, w^2, 1, 0, w, 1, 1, 0, w^2, w, w, 0, w, w^2, 1, w^2, w, w^2, w^2, w, 1, w, w, w, w, w, w, 1, w, w^2, 0, w, 0, w, w^2, 1, 0, 0 ]
where w:=Root(x^2 + x + 1)[1,1];
[2]:  [136, 9, 89] Linear Code over GF(2^2)
Puncturing of [1] at { 137 }

```

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

```Lb(136,9) = 88 is found by taking a subcode of:
Lb(136,10) = 88 DaH

Ub(136,9) = 96 follows by a one-step Griesmer bound from:
Ub(39,8) = 24 follows by a one-step Griesmer bound from:
Ub(14,7) = 6 is found by considering shortening to:
Ub(10,3) = 6 GH
```
###### References
DaH: 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. Email, 2002-2003.

GH: P.P. Greenough & R. Hill, Optimal linear codes over GF(4), Discrete Math. 125 (1994) 187-199.

### 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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