## Bounds on the minimum distance of linear codes

### Bounds on linear codes [228,9] over GF(3)

 lower bound: 143 upper bound: 147

### Construction

```Construction of a linear code [228,9,143] over GF(3):
[1]:  [121, 116, 3] "Hamming code (r = 5)" Linear Code over GF(3)
5-th order HammingCode over GF( 3)
[2]:  [121, 5, 81] Cyclic Linear Code over GF(3)
Dual of [1]
[3]:  [119, 5, 79] Linear Code over GF(3)
Puncturing of [2] at { 120 .. 121 }
[4]:  [40, 4, 27] Linear Code over GF(3)
ResidueCode of [3]
[5]:  [38, 4, 25] Linear Code over GF(3)
Puncturing of [4] at { 39 .. 40 }
[6]:  [13, 3, 9] Linear Code over GF(3)
ResidueCode of [5]
[7]:  [11, 3, 7] Linear Code over GF(3)
Puncturing of [6] at { 12 .. 13 }
[8]:  [4, 2, 3] Linear Code over GF(3)
ResidueCode of [7]
[9]:  [1, 1, 1] Cyclic Linear Code over GF(3)
RepetitionCode of length 1
[10]: [224, 7, 143] Quasicyclic of degree 4 Linear Code over GF(3)
QuasiCyclicCode of length 224 with generating polynomials: x^54 + x^53 + 2*x^51 + 2*x^50 + 2*x^49 + 2*x^48 + x^47 + x^46 + x^45 + 2*x^43 + 2*x^41 + 2*x^40
+ x^39 + x^38 + 2*x^36 + 2*x^34 + 2*x^32 + x^31 + 2*x^30 + x^29 + 2*x^28 + x^27 + x^26 + 2*x^24 + 2*x^23 + 2*x^21 + x^18 + 2*x^16 + 2*x^15 + 2*x^14 + 2*x^13 +
x^10 + x^9 + x^7 + x^5,  x^52 + x^51 + x^50 + 2*x^48 + x^47 + x^46 + 2*x^43 + x^42 + x^39 + 2*x^38 + x^35 + x^32 + x^30 + x^29 + x^28 + 2*x^27 + x^26 + 2*x^25 +
x^23 + 2*x^21 + 2*x^20 + x^19 + 2*x^17 + x^16 + 2*x^13 + x^12 + x^11 + 2*x^10 + 2*x^9 + x^8 + x^7 + x^6 + 2*x^5 + 2*x^3 + x,  x^54 + 2*x^53 + 2*x^52 + x^51 +
2*x^50 + x^49 + x^47 + 2*x^46 + x^45 + 2*x^44 + x^42 + 2*x^41 + x^40 + 2*x^39 + 2*x^38 + x^37 + 2*x^35 + 2*x^34 + 2*x^33 + x^32 + x^31 + 2*x^30 + 2*x^29 +
2*x^27 + 2*x^24 + x^23 + 2*x^22 + x^21 + x^20 + x^19 + 2*x^18 + x^17 + 2*x^16 + 2*x^15 + 2*x^10 + x^9 + x^8 + 2*x^6 + x^3 + 2*x^2 + 1,  2*x^55 + x^54 + 2*x^53 +
2*x^52 + x^50 + 2*x^49 + x^48 + x^47 + 2*x^46 + 2*x^45 + x^44 + 2*x^43 + 2*x^42 + 2*x^41 + 2*x^40 + x^39 + 2*x^38 + x^36 + x^35 + x^34 + x^31 + 2*x^29 + 2*x^27
+ x^26 + 2*x^25 + x^23 + x^22 + 2*x^21 + x^20 + 2*x^17 + x^16 + 2*x^15 + 2*x^13 + x^9 + x^8 + x^6 + x^5 + 2*x^4 + 2*x^2 + 2*x + 2
[11]: [224, 8, 141] Quasicyclic of degree 4 Linear Code over GF(3)
QuasiCyclicCode of length 224 stacked to height 2 with generating polynomials: x^55 + 2*x^54 + 2*x^53 + x^52 + x^51 + 2*x^50 + x^49 + 2*x^47 + 2*x^46 +
x^45 + x^44 + x^43 + x^42 + 2*x^40 + 2*x^39 + 2*x^38 + 2*x^37 + 2*x^35 + 2*x^34 + 2*x^33 + 2*x^32 + x^31 + x^30 + x^28 + 2*x^26 + 2*x^25 + 2*x^22 + x^20 +
2*x^19 + 2*x^18 + x^13 + 2*x^12 + 2*x^11 + 2*x^10 + 2*x^9 + x^8 + 2*x^7 + 2*x^6 + 2*x^5 + 2*x^4 + x,  2*x^55 + x^54 + x^51 + x^50 + x^49 + x^48 + x^45 + x^43 +
x^42 + 2*x^39 + x^38 + 2*x^36 + 2*x^35 + 2*x^33 + 2*x^31 + 2*x^30 + x^29 + x^28 + 2*x^27 + x^25 + x^24 + x^19 + x^18 + x^16 + x^13 + x^12 + 2*x^11 + x^9 + 2*x^8
+ 2*x^7 + x^6 + 2*x^5 + x^4 + 2*x^3 + 2*x^2,  x^55 + x^54 + 2*x^53 + x^52 + x^50 + x^49 + x^45 + x^44 + 2*x^43 + 2*x^41 + 2*x^40 + x^38 + 2*x^35 + 2*x^34 +
2*x^33 + x^32 + x^31 + 2*x^30 + 2*x^28 + x^27 + x^26 + x^24 + 2*x^23 + x^22 + x^21 + 2*x^20 + 2*x^19 + 2*x^18 + x^17 + x^16 + 2*x^14 + 2*x^11 + x^10 + 2*x^9 +
2*x^8 + x^4 + x^3 + 2*x,  2*x^55 + 2*x^54 + 2*x^52 + x^51 + x^50 + 2*x^49 + x^47 + 2*x^44 + 2*x^42 + 2*x^41 + x^40 + 2*x^38 + 2*x^37 + x^36 + x^34 + x^31 +
2*x^29 + 2*x^28 + x^27 + x^26 + x^24 + 2*x^23 + 2*x^22 + x^21 + 2*x^19 + x^16 + x^14 + x^13 + 2*x^12 + x^10 + x^9 + 2*x^8 + 2*x^6 + 2*x^3 + x + 1,  x^54 +
2*x^53 + 2*x^52 + x^50 + x^48 + x^47 + x^45 + 2*x^44 + 2*x^43 + x^40 + 2*x^38 + 2*x^37 + x^35 + 2*x^34 + x^33 + x^32 + x^30 + 2*x^27 + x^26 + 2*x^23 + x^22 +
2*x^21 + x^20 + x^19 + 2*x^18 + x^17 + 2*x^14 + 2*x^13 + x^12 + 2*x^11 + 2*x^8 + x^7 + x^5 + x^4 + 2*x^3 + x^2 + 2*x + 2,  x^55 + 2*x^53 + 2*x^51 + 2*x^49 +
2*x^46 + x^45 + x^44 + 2*x^43 + 2*x^42 + 2*x^40 + 2*x^39 + x^38 + 2*x^35 + 2*x^34 + 2*x^32 + 2*x^31 + x^29 + 2*x^28 + x^26 + 2*x^25 + x^24 + 2*x^23 + x^22 +
2*x^21 + x^20 + x^19 + 2*x^18 + 2*x^15 + 2*x^14 + x^13 + 2*x^12 + 2*x^11 + x^9 + x^8 + 2*x^7 + 2*x^6 + x^5 + 2*x^4 + 2*x^3 + x^2 + 2,  x^55 + 2*x^54 + x^53 +
2*x^52 + 2*x^51 + 2*x^49 + 2*x^48 + 2*x^45 + 2*x^44 + x^43 + x^42 + 2*x^41 + 2*x^40 + x^39 + 2*x^36 + x^35 + x^33 + 2*x^32 + 2*x^31 + 2*x^30 + 2*x^29 + 2*x^27 +
x^26 + 2*x^25 + x^24 + x^23 + x^21 + x^20 + x^17 + x^16 + 2*x^15 + 2*x^14 + x^13 + x^12 + 2*x^11 + x^8 + 2*x^7 + 2*x^5 + x^4 + x^3 + x^2 + x,  x^54 + x^53 +
x^51 + 2*x^50 + x^49 + 2*x^47 + x^46 + 2*x^45 + 2*x^43 + 2*x^41 + x^40 + 2*x^39 + 2*x^36 + x^32 + x^31 + 2*x^30 + 2*x^29 + 2*x^28 + x^27 + x^24 + 2*x^22 + x^20
+ 2*x^19 + 2*x^17 + x^16 + 2*x^15 + x^14 + 2*x^13 + 2*x^11 + x^10 + x^9 + 2*x^8 + x^7 + x^6 + x^5 + 2*x^2 + 2*x + 2
[12]: [224, 9, 140] Quasicyclic of degree 4 Linear Code over GF(3)
QuasiCyclicCode of length 224 stacked to height 2 with generating polynomials: 2*x^54 + 2*x^49 + 2*x^48 + 2*x^47 + x^45 + 2*x^44 + 2*x^43 + x^40 + 2*x^39 +
2*x^36 + x^35 + 2*x^32 + 2*x^29 + 2*x^27 + 2*x^26 + 2*x^25 + x^24 + 2*x^23 + x^22 + 2*x^20 + x^18 + x^17 + 2*x^16 + x^14 + 2*x^13 + x^10 + 2*x^9 + 2*x^8 + x^7 +
x^6 + 2*x^5 + 2*x^4 + 2*x^3 + x^2 + 1,  x^55 + x^54 + 2*x^53 + 2*x^52 + 2*x^51 + x^50 + 2*x^49 + 2*x^48 + x^47 + x^46 + x^44 + 2*x^43 + x^42 + 2*x^41 + x^40 +
x^39 + x^37 + x^35 + x^34 + x^32 + x^30 + 2*x^29 + x^28 + x^27 + 2*x^26 + x^24 + 2*x^22 + x^20 + x^19 + 2*x^18 + 2*x^17 + 2*x^16 + 2*x^14 + 2*x^12 + x^11 + x^9
+ x^7 + 2*x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 2,  x^54 + x^52 + 2*x^51 + 2*x^50 + 2*x^47 + 2*x^46 + x^45 + x^39 + x^38 + 2*x^37 + x^34 + 2*x^33 + 2*x^32 +
2*x^30 + x^28 + 2*x^27 + 2*x^26 + 2*x^25 + 2*x^24 + x^22 + 2*x^21 + x^18 + x^17 + 2*x^15 + 2*x^13 + x^11 + 2*x^10 + 2*x^7 + 2*x^6 + x^4 + 2*x^3 + x^2 + 2*x + 2,
x^55 + 2*x^52 + x^49 + x^47 + 2*x^46 + x^45 + 2*x^44 + 2*x^43 + x^42 + x^41 + x^40 + x^39 + x^37 + x^34 + x^31 + 2*x^29 + x^25 + x^24 + x^23 + x^18 + x^16 +
2*x^15 + 2*x^14 + 2*x^12 + x^7 + 2*x^6 + x^5 + 2*x,  x^54 + x^53 + x^52 + x^50 + 2*x^49 + x^48 + x^46 + 2*x^45 + 2*x^43 + 2*x^42 + x^41 + x^40 + 2*x^39 + 2*x^38
+ 2*x^37 + 2*x^36 + 2*x^35 + x^33 + 2*x^32 + 2*x^31 + x^30 + 2*x^29 + 2*x^25 + x^21 + x^17 + x^16 + x^15 + 2*x^14 + 2*x^13 + x^11 + 2*x^10 + x^9 + 2*x^8 + x^7 +
x^6 + 2*x^5 + 2*x^4 + x^3 + x + 1,  2*x^54 + 2*x^53 + 2*x^51 + 2*x^49 + 2*x^42 + 2*x^41 + x^39 + x^38 + x^37 + x^36 + 2*x^35 + 2*x^34 + 2*x^33 + x^31 + x^29 +
x^28 + 2*x^27 + 2*x^26 + x^24 + x^22 + x^20 + 2*x^19 + x^18 + 2*x^17 + x^16 + 2*x^15 + 2*x^14 + x^12 + x^11 + x^9 + 2*x^6 + x^4 + x^3 + x^2 + x,  2*x^53 +
2*x^50 + x^49 + x^48 + x^47 + x^45 + x^44 + 2*x^43 + 2*x^42 + 2*x^41 + x^40 + 2*x^39 + x^38 + 2*x^37 + 2*x^36 + 2*x^35 + x^34 + x^30 + 2*x^28 + x^27 + 2*x^25 +
x^23 + x^22 + 2*x^20 + 2*x^16 + 2*x^15 + x^14 + 2*x^13 + 2*x^12 + 2*x^11 + 2*x^10 + 2*x^9 + x^8 + 2*x^7 + 2*x^6 + x^5 + x^3 + 2*x^2 + x + 1,  x^55 + 2*x^54 +
x^53 + 2*x^52 + x^50 + 2*x^49 + 2*x^48 + x^47 + x^46 + 2*x^45 + x^43 + x^41 + x^40 + 2*x^39 + 2*x^37 + x^35 + x^34 + 2*x^33 + 2*x^32 + x^31 + x^30 + 2*x^28 +
2*x^27 + 2*x^26 + 2*x^25 + 2*x^24 + x^21 + 2*x^20 + 2*x^19 + x^17 + x^16 + 2*x^15 + x^14 + 2*x^13 + x^11 + x^10 + x^9 + x^8 + 2*x^7 + x^5 + 2*x^4 + 2*x^3 + 2
[13]: [229, 9, 144] Linear Code over GF(3)
ConstructionXX using [12] [11] [10] [9] and [8]
[14]: [228, 9, 143] Linear Code over GF(3)
Puncturing of [13] at { 229 }

```

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

```Lb(228,9) = 140 is found by taking a subcode of:
Lb(228,10) = 140 MSY

Ub(228,9) = 147 follows by a one-step Griesmer bound from:
Ub(80,8) = 49 follows by a one-step Griesmer bound from:
Ub(30,7) = 16 is found by considering shortening to:
Ub(29,6) = 16 is found by considering truncation to:
Ub(28,6) = 15 HHM
```
###### References
HHM: N. Hamada, T. Helleseth, H.M. Martinsen & Ø. Ytrehus, There is no ternary [28,6,16] code

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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