## Bounds on the minimum distance of linear codes

### Bounds on linear codes [156,9] over GF(4)

 lower bound: 104 upper bound: 111

### Construction

```Construction of a linear code [156,9,104] over GF(4):
[1]:  [162, 9, 110] Linear Code over GF(2^2)
Construction from a stored generator matrix:

[ 1, 0, 0, 0, 0, 0, 0, 0, 0, w^2, 0, w^2, 0, 1, w^2, w^2, w, 0, 1, w, 1, 0, 0, w, w^2, 0, 1, 1, 0, 1, w, 1, w, w^2, 1, 1, 0, 1, w, w, 1, 0, 1, 0, 0, 0, 1, w^2,
0, w^2, w^2, w^2, 0, w, 0, 0, w^2, 0, w, w^2, w, 1, 1, w^2, w^2, 1, w^2, w, w^2, 0, 1, w^2, 0, 1, w^2, w^2, w, w, w, 1, w^2, 0, 1, w, 1, 1, 0, w, w, 1, 0, w^2,
w, 1, 1, 1, w^2, w^2, w, 0, 0, w^2, w, 0, w^2, 1, w^2, w, w^2, w, w, 1, w^2, 1, 0, w^2, 0, 0, w^2, 1, 0, w^2, w^2, w^2, 0, w^2, 0, 0, 0, w, 0, w, 1, 1, w, 0, w,
w, w, w^2, 1, w, 1, w, w, w, 0, w^2, 1, 0, 0, w, 1, w, w, 0, 1, w^2, w^2, w, 0, w^2 ]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, w^2, 0, w^2, 0, 1, w^2, w^2, w, 1, 1, w, 1, 0, 0, w, w^2, 0, 1, 1, 0, 1, w, 1, w, w^2, 1, 1, 0, 1, w, w, 1, 0, 1, 0, 0, 0, 1,
w^2, 0, w^2, w^2, w^2, 0, w^2, 0, 0, w^2, 0, w, w^2, w, 1, 1, w^2, w^2, 1, w^2, w, w^2, 0, 1, 0, 0, 1, w^2, w^2, w, w, w, 1, w^2, 0, 1, w, 1, 1, 0, w, w, w, 0,
w^2, w, 1, 1, 1, w^2, w^2, w, 0, 0, w^2, w, 0, w^2, 1, w^2, w^2, w^2, w, w, 1, w^2, 1, 0, w^2, 0, 0, w^2, 1, 0, w^2, w^2, w^2, 0, w, 0, 0, 0, w, 0, w, 1, 1, w,
0, w, w, w, w^2, 1, w, 1, 0, w, w, 0, w^2, 1, 0, 0, w, 1, w, w, 0, 1, w^2, w^2, w, 0 ]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0, w^2, 0, 0, 0, w, w^2, w, 1, w^2, 1, w^2, 0, w, 1, w, w^2, w, w, 1, 1, 0, w, 0, 0, w, w^2, w, 1, 0, w, w^2, w^2, w, 0, 0, 1, 0, 1,
w^2, 1, 0, w^2, 0, w^2, 1, w, w^2, w^2, 0, 1, w^2, 0, w, w^2, w, w, w, 0, w^2, 0, w, w, w^2, w^2, 1, w^2, w, 1, 1, 0, w^2, 1, 1, w, w, 0, w^2, 1, w^2, w, w^2,
1, 1, w, w, w^2, 0, w, w, 1, w^2, w, w^2, w, w^2, 1, 1, 0, w^2, 1, 1, 1, w^2, 1, 0, w^2, w, 0, w^2, w^2, 1, w^2, w, w^2, 0, w^2, 0, w^2, w, 0, w, 0, 0, 1, w^2,
w^2, 1, 0, w, 0, 1, w^2, 1, 0, 0, 0, w, w, 1, 1, w^2, 1, w, 1, 0, w^2, w, w^2, w^2, w, 1, w^2, 1 ]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, w^2, 0, 0, 0, w, w^2, w, 1, w^2, 1, w^2, 0, w, 1, w, w^2, w, w, 1, 1, 0, w, 0, 0, w, w^2, 0, 1, 0, w, w^2, w^2, w, 0, 0, 1, 0,
1, w^2, 1, 0, w^2, 0, w^2, w^2, w, w^2, w^2, 0, 1, w^2, 0, w, w^2, w, w, w, 0, w^2, 0, w, w, 1, w^2, 1, w^2, w, 1, 1, 0, w^2, 1, 1, w, w, 0, w^2, 1, w^2, w, 0,
1, 1, w, w, w^2, 0, w, w, 1, w^2, w, w^2, w, w^2, 1, 1, 0, 0, 1, 1, 1, w^2, 1, 0, w^2, w, 0, w^2, w^2, 1, w^2, w, w^2, 0, w^2, w, w^2, w, 0, w, 0, 0, 1, w^2,
w^2, 1, 0, w, 0, 1, w^2, 1, 0, 1, 0, w, w, 1, 1, w^2, 1, w, 1, 0, w^2, w, w^2, w^2, w, 1, w^2 ]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, w, 0, w, 1, w^2, 0, w, 1, 0, w^2, w^2, 0, 1, 0, w, 1, 0, w, w^2, w, w, 1, 1, w, 0, w, w, w, w^2, 0, w^2, 1, w, 0, 0, w^2, 1,
1, w, 0, 1, w^2, w^2, 1, w^2, w^2, w^2, 0, 1, w, 1, w, w^2, w, 0, w^2, 1, 1, w^2, w, w^2, w^2, w^2, w, 0, 0, 1, w, w^2, 1, w^2, w^2, w, 0, 0, 0, 0, w, 1, w^2,
1, w, w^2, 0, 0, w, 1, 1, w, 1, w, 1, w^2, w^2, 1, 0, w, w, w^2, w, w^2, 0, 1, 1, 1, w^2, w, 1, 1, w^2, 0, w, w^2, w^2, 1, 0, w, w^2, 1, 0, 1, w, w, w, w^2, 0,
w, w^2, w^2, w, w, 1, w, w^2, w, 0, w^2, 0, 1, 1, 0, w^2, 1, w, 0, 1, w^2, w, 0, w, 0 ]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0, w^2, 1, w^2, w, 1, 1, w, 1, 0, w, w^2, 1, w^2, w^2, w, w, 0, w^2, 0, 0, w^2, 1, w^2, 0, w, 0, w^2, 1, w^2, 0, 0, w, 0, w, 1, w, 0,
1, 0, 1, w, 1, w^2, 1, 1, w, 1, 0, w^2, 1, w^2, w^2, w^2, 0, 1, 0, w^2, w^2, 1, w, 1, w, 1, w, w, 0, 1, w, w, w^2, w^2, 0, 1, w, 1, w^2, 1, 0, w, w, w^2, 1, 0,
w^2, w^2, w, 1, w^2, 1, w^2, 1, w, w, 0, 1, 0, w, w, w, w, 0, 1, w^2, 0, 1, 1, w, 1, w^2, 1, 0, 1, 0, w^2, 1, w^2, 0, 0, 0, w, 1, 1, w, 0, w^2, 0, w, 1, w, 0,
0, w, 0, w^2, w^2, w, 1, w, w^2, w, 0, 1, w^2, 1, 1, w^2, w, 1, w, 0, 0, 0, 1 ]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, w^2, 1, w^2, w, 1, 1, w, 1, w^2, w, w^2, 1, w^2, w^2, w, w, 0, w^2, 0, 0, w^2, 1, w^2, 0, w, 0, 1, 1, w^2, 0, 0, w, 0, w, 1, w,
0, 1, 0, 1, w, 1, w^2, 1, 0, w, 1, 0, w^2, 1, w^2, w^2, w^2, 0, 1, 0, w^2, w^2, 1, w, 1, w, w^2, w, w, 0, 1, w, w, w^2, w^2, 0, 1, w, 1, w^2, 1, 0, w, w, w^2,
1, 0, w^2, w^2, w, 1, w^2, 1, w^2, 1, w, w, 0, 1, 0, w, w, 1, w, 0, 1, w^2, 0, 1, 1, w, 1, w^2, 1, 0, 1, 0, w^2, 1, w^2, w^2, 0, 0, w, 1, 1, w, 0, w^2, 0, w, 1,
w, 0, 0, w, 0, w^2, w, w, 1, w, w^2, w, 0, 1, w^2, 1, 1, w^2, w, 1, w, 0, 0, 0 ]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, w, 1, 1, w^2, 0, w, w, w, 0, 0, w^2, 1, 0, w, w, 0, w, w^2, w, w^2, 1, w, w, w, 0, w^2, w^2, w, 0, w, 0, 0, 0, w, 1, 0, 1, 1,
1, 0, w^2, 1, 0, 1, 0, w^2, 1, w^2, w, w, 1, 1, w, 1, w^2, 1, 0, w, 1, 0, 0, 1, 1, w^2, w^2, w^2, w, 1, 0, w, w^2, w, w, 0, w^2, w^2, w, w^2, 0, w^2, w, w, w,
1, 1, w^2, 0, 0, 1, w^2, 0, 1, w, 1, w^2, 1, 1, w^2, w, 1, w, 0, 1, 0, 0, 1, w, 0, 1, 1, 1, 0, 1, w^2, 0, 0, w^2, 0, w^2, w, w, w^2, 0, w^2, w^2, w^2, 1, w,
w^2, w, w^2, 0, w^2, 0, 1, w, 0, 0, w^2, w, w^2, w^2, 0, w, 1, 1, w^2, 0, 1, 0, 1 ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, w, 1, 1, w^2, 0, w, w^2, w, 0, 0, w^2, 1, 0, w, w, 0, w, w^2, w, w^2, 1, w, w, w, w, w^2, w^2, w, 0, w, 0, 0, 0, w, 1, 0,
1, 1, 1, 0, w^2, 1, 0, 1, 0, w^2, 1, w^2, w, w, 1, 1, w, 1, w^2, 1, 0, w, 1, 0, w, 1, 1, w^2, w^2, w^2, w, 1, 0, w, w^2, w, w, 0, w^2, w^2, w, w^2, 1, w^2, w,
w, w, 1, 1, w^2, 0, 0, 1, w^2, 0, 1, w, 1, w^2, 1, w^2, w^2, w, 1, w, 0, 1, 0, 0, 1, w, 0, 1, 1, 1, 0, 1, w^2, 0, 0, w^2, 0, w^2, w, w, w^2, 0, w^2, w^2, w^2,
1, w, w^2, w, w^2, 0, w^2, 0, 1, w, 0, 0, w^2, w, w^2, w^2, 0, w, 1, 1, w^2, 0, 1, 0 ] where w:=Root(x^2 + x + 1)[1,1];
[2]:  [156, 9, 104] Linear Code over GF(2^2)
Puncturing of [1] at { 157 .. 162 }

```

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

```Lb(156,9) = 104 is found by truncation of:
Lb(162,9) = 110 MSY

Ub(156,9) = 111 is found by considering shortening to:
Ub(154,7) = 111 DM3
```
###### References
DM3: R. N. Daskalov & E. Metodieva, Bounds on minimum length for quaternary linear codes in dimensions six and seven, Mathematics and Education in Mathematics, Sofia, (1994) 156-161.

MSY: T. Maruta, M. Shinohara, F. Yamane, K. Tsuji, E. Takata, H. Miki & R. Fujiwara, New linear codes from cyclic or generalized cyclic codes by puncturing, to appear in Proc. 10th International Workshop on Algebraic and Combinatorial Coding Theory(ACCT-10) in Zvenigorod, Russia, 2006.

### 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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