## Bounds on the minimum distance of linear codes

### Bounds on linear codes [169,23] over GF(3)

 lower bound: 78 upper bound: 94

### Construction

Construction type: GraXX

```Construction of a linear code [169,23,78] over GF(3):
[1]:  [121, 116, 3] "Hamming code (r = 5)" Linear Code over GF(3)
5-th order HammingCode over GF( 3)
[2]:  [118, 113, 3] Linear Code over GF(3)
Shortening of [1] at { 119 .. 121 }
[3]:  [40, 36, 3] Linear Code over GF(3)
Construction B of [2]
[4]:  [37, 33, 3] Linear Code over GF(3)
Shortening of [3] at { 38 .. 40 }
[5]:  [13, 10, 3] Linear Code over GF(3)
Construction B of [4]
[6]:  [7, 4, 3] Linear Code over GF(3)
Shortening of [5] at { 8 .. 13 }
[7]:  [2, 1, 2] Cyclic Linear Code over GF(3)
RepetitionCode of length 2
[8]:  [160, 19, 76] Quasicyclic of degree 2 Linear Code over GF(3)
QuasiCyclicCode of length 160 with generating polynomials: 2*x^79 + x^78 + x^77 + x^73 + 2*x^72 + x^71 + x^68 + x^67 + x^65 + x^62 + 2*x^57 + x^55 + 2*x^54 + x^53 + 2*x^52 + x^51 + x^46 + 2*x^45 + 2*x^42 + x^39 + x^37 + x^36 + 2*x^35 + x^34 + 2*x^32 + 2*x^31 + x^30 + x^27 + x^25 + 2*x^24 + 2*x^23 + 2*x^22 + x^20 + x^19 + x^7,  x^79 + 2*x^78 + 2*x^77 + x^76 + x^75 + x^74 + 2*x^73 + x^72 + x^71 + 2*x^70 + 2*x^66 + 2*x^65 + x^63 + 2*x^61 + 2*x^57 + 2*x^56 + 2*x^54 + x^53 + x^52 + x^50 + 2*x^49 + x^48 + 2*x^46 + x^43 + 2*x^42 + 2*x^41 + x^40 + 2*x^36 + 2*x^35 + 2*x^34 + x^33 + x^32 + 2*x^30 + 2*x^29 + x^28 + 2*x^27 + x^26 + x^24 + 2*x^22 + 2*x^20 + 2*x^19 + 2*x^18 + x^17 + 2*x^16 + 2*x^15 + x^14 + x^13 + 2*x^11 + x^9 + x^8 + x^7 + x^6 + x^5 + x^3 + x + 2
[9]:  [160, 22, 75] Quasicyclic of degree 2 Linear Code over GF(3)
QuasiCyclicCode of length 160 with generating polynomials: x^77 + x^76 + 2*x^74 + 2*x^73 + x^72 + x^71 + x^70 + 2*x^68 + 2*x^66 + x^65 + 2*x^64 + x^59 + x^56 + x^54 + 2*x^53 + x^52 + 2*x^50 + x^49 + 2*x^48 + x^46 + 2*x^45 + 2*x^44 + x^40 + x^39 + x^38 + 2*x^37 + x^36 + 2*x^34 + x^33 + 2*x^31 + 2*x^30 + x^28 + 2*x^27 + x^23 + x^22 + x^7,  x^77 + x^76 + 2*x^75 + x^74 + 2*x^71 + x^70 + 2*x^68 + x^67 + 2*x^65 + x^64 + 2*x^62 + 2*x^61 + 2*x^60 + 2*x^59 + 2*x^58 + 2*x^57 + 2*x^56 + x^53 + x^52 + 2*x^51 + x^50 + x^49 + x^48 + x^47 + 2*x^46 + 2*x^45 + 2*x^43 + 2*x^41 + 2*x^40 + 2*x^39 + 2*x^38 + 2*x^37 + 2*x^36 + x^33 + x^32 + x^31 + x^30 + x^29 + x^27 + 2*x^26 + x^25 + 2*x^24 + x^22 + x^15 + 2*x^14 + x^12 + x^11 + x^10 + 2*x^8 + x^7 + x^5 + 2*x^3 + 2*x + 1
[10]: [160, 23, 73] Quasicyclic of degree 2 Linear Code over GF(3)
QuasiCyclicCode of length 160 with generating polynomials: 2*x^78 + x^76 + x^75 + x^74 + x^73 + x^72 + 2*x^70 + x^69 + 2*x^68 + 2*x^67 + 2*x^65 + 2*x^64 + 2*x^63 + x^61 + 2*x^60 + 2*x^59 + x^58 + 2*x^57 + x^56 + x^55 + x^54 + 2*x^53 + 2*x^52 + 2*x^51 + x^49 + 2*x^48 + 2*x^47 + 2*x^46 + x^45 + x^44 + 2*x^42 + 2*x^41 + 2*x^40 + x^39 + 2*x^38 + x^37 + x^36 + x^35 + 2*x^32 + x^30 + x^28 + x^27 + 2*x^26 + 2*x^25 + 2*x^24 + 2*x^23 + x^13,  2*x^79 + 2*x^78 + x^76 + 2*x^75 + 2*x^74 + x^72 + x^71 + x^70 + 2*x^68 + 2*x^67 + x^66 + 2*x^64 + 2*x^62 + 2*x^60 + 2*x^58 + 2*x^56 + x^53 + x^50 + 2*x^49 + x^47 + x^46 + x^45 + x^44 + 2*x^43 + x^42 + x^40 + x^39 + x^38 + x^36 + 2*x^33 + 2*x^32 + 2*x^31 + 2*x^30 + x^29 + x^28 + 2*x^27 + 2*x^25 + x^24 + x^23 + x^22 + 2*x^19 + 2*x^18 + 2*x^16 + 2*x^15 + x^12 + 2*x^11 + x^10 + 2*x^9 + x^8 + 2*x^7 + 2*x^4 + x^3 + 2*x
[11]: [169, 23, 78] Linear Code over GF(3)
ConstructionXX using [10] [9] [8] [7] and [6]

```

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

```Lb(169,23) = 74 is found by truncation of:
Lb(176,23) = 81 DaH

Ub(169,23) = 94 is found by considering shortening to:
Ub(166,20) = 94 Da2
```
###### References
Da2: R.N. Daskalov, The linear programming bound for ternary linear codes, p. 423 in: IEEE International Symposium on Information Theory, Trondheim, 1994.

DaH: Rumen Daskalov & Plamen Hristov, New One-Generator Quasi-Cyclic Codes over GF(7), preprint, Oct 2001. R. Daskalov & P Hristov, New One-Generator Quasi-Twisted Codes over GF(5), (preprint) Oct. 2001. R. Daskalov & P Hristov, New Quasi-Twisted Degenerate Ternary Linear Codes, preprint, Nov 2001. Email, 2002-2003.

### 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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This page is maintained by Markus Grassl (codes@codetables.de). Last change: 30.12.2011