## Bounds on the minimum distance of linear codes

### Bounds on linear codes [68,11] over GF(5)

 lower bound: 40 upper bound: 46

### Construction

Construction type: EricChen

```Construction of a linear
code [68,11,40] over GF(5):
:  [2, 1, 2] Cyclic Linear Code over GF(5)
RepetitionCode of length 2
:  [66, 10, 40] Quasicyclic of degree 3 Linear Code over GF(5)
QuasiCyclicCode of length 66 with generating polynomials: x^21 + 2*x^20 +
4*x^19 + x^18 + 4*x^17 + 2*x^15 + 4*x^13 + 4*x^12 + 4*x^11 + 3*x^10 + x^9,
4*x^21 + 2*x^20 + 3*x^19 + 3*x^18 + 3*x^17 + x^16 + 4*x^15 + x^14 + 3*x^13 +
2*x^12 + 3*x^11 + 4*x^10 + 3*x^9 + x^8 + 3*x^7 + 4*x^6 + x^4 + x^3 + x^2 + 3*x,
4*x^21 + 2*x^20 + 3*x^19 + 4*x^18 + 4*x^17 + 2*x^15 + x^13 + 4*x^12 + x^11 +
x^10 + 4*x^9 + x^8 + x^7 + x^6 + x^2 + 1
:  [66, 11, 38] Quasicyclic of degree 3 Linear Code over GF(5)
QuasiCyclicCode of length 66 with generating polynomials: x^21 + 2*x^20 +
x^19 + 4*x^18 + 3*x^17 + x^16 + 2*x^15 + 3*x^14 + x^13 + x^12 + 2*x^11 + x^5,
x^21 + 3*x^20 + 3*x^19 + 2*x^18 + 3*x^17 + 2*x^16 + 2*x^15 + x^14 + 2*x^13 +
2*x^12 + x^11 + 2*x^10 + 2*x^9 + 4*x^8 + 2*x^6 + 4*x^5 + 4*x^3 + 4*x^2 + x + 1,
2*x^21 + x^20 + x^19 + 3*x^18 + 3*x^17 + 2*x^16 + 4*x^14 + x^13 + 4*x^12 + x^11
+ 4*x^10 + x^9 + 2*x^8 + 2*x^7 + 3*x^5 + 2*x^4 + 2*x^3 + x^2 + 4*x + 2
:  [68, 11, 40] Linear Code over GF(5)
ConstructionX using   and 

```

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

```Lb(68,11) = 39 is found by truncation of:
Lb(69,11) = 40 Ma

Ub(68,11) = 46 is found by considering shortening to:
Ub(66,9) = 46 is found by considering truncation to:
Ub(65,9) = 45 is found by construction B:
[consider deleting the (at most) 6 coordinates of a word in the dual]
```
###### References
Ma: T. Maruta, On the nonexistence of linear codes attaining the Griesmer bound, Geom. Dedicata 60 (1996) 1-7. T. Maruta, On the nonexistence of linear codes of dimension four attaining the Griesmer bound, pp. 117-120 in: Optimal codes and related topics, Proc. Workshop Sozopol, Bulgaria, 1995. T. Maruta, The nonexistence of [116,5,85]_4 codes and [187,5,139]_4 codes, Proc. 2nd International Workshop on Optimal Codes and Related Topics in Sozopol (1998), pp. 168-174. T. Maruta & M. Fukui, On the nonexistence of some linear codes of dimension 4 over GF(5), preprint, 1995. T. Maruta, M. Takenaka, M. Shinohara, K. Masuda & S. Kawashima, Constructing new linear codes over small fields, preprint 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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