## Bounds on the minimum distance of linear codes

### Bounds on linear codes [135,12] over GF(3)

 lower bound: 75 upper bound: 81

### Construction

Construction type: Kohnert

```Construction of a linear code [135,12,75] over GF(3):
[1]:  [1, 1, 1] Cyclic Linear Code over GF(3)
RepetitionCode of length 1
[2]:  [132, 11, 73] Quasicyclic of degree 12 Linear Code over GF(3)
QuasiCyclicCode of length 132 with generating polynomials: x^3,  2*x^10 + 2*x^9 + 2*x^8 + 2*x^4 + x^2 + 2*x,  2*x^6 + 2*x^5 + x^3 + x,  2*x^10 + x^8 + x^7
+ 2*x^6 + x^3 + x^2 + 2*x + 2,  x^9 + x^8 + 2*x^5 + x^4 + x^2 + 2,  x^10 + x^9 + x^6 + 2*x^5 + 2*x^3 + x^2 + 2,  2*x^10 + 2*x^9 + 2*x^7 + x^6 + x^5 + 2*x^3 +
2*x^2,  2*x^10 + x^8 + x^7 + x^5 + 2*x^4 + x^3 + 2*x + 2,  x^9 + x^8 + x^7 + x^6 + 2*x^5 + x^4 + x + 2,  2*x^10 + x^9 + x^8 + x^7 + 2*x^6 + x^2 + x + 1,  2*x^10
+ x^9 + x^8 + x^7 + 2*x^5 + 2*x^4 + 2*x^2,  x^10 + 2*x^8 + x^7 + 2*x^6 + x^4 + x^3 + 2*x^2 + 2*x + 1
[3]:  [132, 11, 73] Quasicyclic of degree 12 Linear Code over GF(3)
QuasiCyclicCode of length 132 with generating polynomials: x^8,  2*x^9 + x^7 + 2*x^6 + 2*x^4 + 2*x^3 + 2*x^2,  x^10 + 2*x^9 + 2*x^7 + 2*x^5 + 2*x^4 + 2*x^3
+ 2*x^2 + 2*x + 1,  2*x^10 + 2*x^9 + x^6 + x^5 + x^4 + 2*x^3 + 1,  x^10 + 2*x^8 + 2*x^6 + x^5 + 2*x^4 + 2*x + 2,  x^9 + 2*x^7 + x^6 + 2*x^4 + 2*x^3 + x^2 + x +
2,  2*x^9 + x^8 + x^7 + 2*x^6 + 2*x^5 + x^4 + x^3 + 2*x^2 + x,  2*x^10 + 2*x^8 + x^7 + x^3 + 2*x^2 + 2*x + 1,  2*x^10 + x^9 + x^6 + 2*x^5 + x^3 + x^2 + x + 1,
x^7 + x^6 + x^5 + 2*x^4 + x^3 + x^2 + x + 2,  x^10 + x^9 + 2*x^8 + x^7 + 2*x^6 + 2*x^5 + x^4 + 2,  x^10 + 2*x^9 + 2*x^8 + 2*x^5 + 2*x^4 + x^3 + 2*x
[4]:  [132, 11, 73] Quasicyclic of degree 12 Linear Code over GF(3)
QuasiCyclicCode of length 132 with generating polynomials: 2*x^10 + 1,  x^10 + x^9 + x^8 + 2*x^7 + x^4 + 2*x + 1,  x^10 + x^8 + 2*x^7 + 2*x^6 + 2*x^5 +
2*x^4 + x^3 + 2*x^2,  2*x^10 + 2*x^8 + 2*x^7 + 2*x^4 + 2*x,  x^9 + x^8 + 2*x^5 + x^4 + 2*x^3 + x^2 + x + 1,  x^6 + x^4 + 2*x^3 + 2*x^2 + 2*x,  2*x^10 + 2*x^8 +
x^7 + 2*x^6 + x^4 + x^3 + 2*x^2 + x + 1,  x^8 + x^7 + 2*x^6 + 2*x^5 + x^4 + 2*x^2 + 2*x,  x^9 + x^8 + x^7 + x^6 + x^2 + 2*x + 2,  x^10 + x^7 + 2*x^6 + x^3 +
x^2,  x^10 + 2*x^8 + x^7 + x^6 + 2*x^5 + 2*x^4 + x^3 + x^2 + 2*x,  2*x^10 + x^9 + x^7 + 2*x^3 + 2*x^2 + 2*x + 1
[5]:  [132, 12, 72] Linear Code over GF(3)
QuasiCyclicCode of length 132 stacked to height 2 with generating polynomials: 0,  0,  x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,  x^10
+ x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,  x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,  2*x^10 + 2*x^9 + 2*x^8 + 2*x^7 + 2*x^6
+ 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 2*x + 2,  x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,  2*x^10 + 2*x^9 + 2*x^8 + 2*x^7 + 2*x^6 + 2*x^5 +
2*x^4 + 2*x^3 + 2*x^2 + 2*x + 2,  0,  0,  x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,  2*x^10 + 2*x^9 + 2*x^8 + 2*x^7 + 2*x^6 + 2*x^5 + 2*x^4
+ 2*x^3 + 2*x^2 + 2*x + 2,  x^8,  2*x^9 + x^7 + 2*x^6 + 2*x^4 + 2*x^3 + 2*x^2,  x^10 + 2*x^9 + 2*x^7 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 2*x + 1,  2*x^10 + 2*x^9
+ x^6 + x^5 + x^4 + 2*x^3 + 1,  x^10 + 2*x^8 + 2*x^6 + x^5 + 2*x^4 + 2*x + 2,  x^9 + 2*x^7 + x^6 + 2*x^4 + 2*x^3 + x^2 + x + 2,  2*x^9 + x^8 + x^7 + 2*x^6 +
2*x^5 + x^4 + x^3 + 2*x^2 + x,  2*x^10 + 2*x^8 + x^7 + x^3 + 2*x^2 + 2*x + 1,  2*x^10 + x^9 + x^6 + 2*x^5 + x^3 + x^2 + x + 1,  x^7 + x^6 + x^5 + 2*x^4 + x^3 +
x^2 + x + 2,  x^10 + x^9 + 2*x^8 + x^7 + 2*x^6 + 2*x^5 + x^4 + 2,  x^10 + 2*x^9 + 2*x^8 + 2*x^5 + 2*x^4 + x^3 + 2*x
[6]:  [135, 12, 75] Linear Code over GF(3)
Apply ConstructionXChain to [5] [4] [3] [2] and [1] then apply ConstructionXX using [1] [1]

```

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

```Lb(135,12) = 72 is found by taking a subcode of:
Lb(135,13) = 72 BZ

Ub(135,12) = 81 Gur
```
###### References
BZ: E. L. Blokh & V. V. Zyablov, Coding of generalized concatenated codes, Probl. Inform. Transm. 10 (1974) 218-222.

Gur: Sugi Guritman, Restrictions on the weight distribution of linear codes, Thesis, Techn. Univ. Delft, 2000.

### 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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