## Bounds on the minimum distance of linear codes

### Bounds on linear codes [240,9] over GF(4)

 lower bound: 164 upper bound: 173

### Construction

```Construction of a linear code [240,9,164] over GF(4):
:  [255, 245, 4] Cyclic Linear Code over GF(2^2)
CyclicCode of length 255 with generating polynomial x^10 + x^9 + w*x^8 + w^2*x^7 + w^2*x^6 + w*x^2 + w^2*x + w
:  [255, 10, 176] Cyclic Linear Code over GF(2^2)
Dual of 
:  [240, 10, 164] Linear Code over GF(2^2)
Puncturing of  at { 1, 18, 35, 52, 69, 86, 103, 120, 137, 154, 171, 188, 205, 222, 239 }
:  [240, 9, 164] Linear Code over GF(2^2)
Subcode of 

```

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

```Lb(240,9) = 164 is found by taking a subcode of:
Lb(240,10) = 164 GW1

Ub(240,9) = 173 is found by considering shortening to:
Ub(239,8) = 173 is found by considering truncation to:
Ub(237,8) = 171 DM4
```
###### References
DM4: R. N. Daskalov & E. Metodieva, The Linear Programming Bound for Ternary and Quaternary Linear Codes, preprint, Jan 2002.

GW1: M. Grassl & G. White, New Good Linear Codes by Special Puncturings, ISIT 2004 Chicago USA June 27 - July 2 2004.

### 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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