## Bounds on the minimum distance of linear codes

### Bounds on linear codes [96,9] over GF(4)

 lower bound: 60 upper bound: 66

### Construction

```Construction of a linear code [96,9,60] over GF(4):
[1]:  [3, 1, 3] Cyclic Linear Code over GF(2^2)
RepetitionCode of length 3
[2]:  [12, 6, 6] Linear Code over GF(2^2)
Extend the QRCode over GF(4)of length 11
[3]:  [10, 4, 6] Linear Code over GF(2^2)
Shortening of [2] at { 11 .. 12 }
[4]:  [85, 8, 56] Cyclic Linear Code over GF(2^2)
CyclicCode of length 85 with generating polynomial x^77 + w^2*x^76 + w*x^75 + w^2*x^74 + x^73 + w^2*x^72 + x^71 + w^2*x^70 + x^69 + x^68 + w^2*x^67 +
w*x^64 + w^2*x^63 + w*x^61 + w*x^60 + w*x^58 + x^57 + x^56 + x^54 + w*x^53 + w^2*x^52 + w*x^51 + w*x^50 + x^47 + w*x^46 + x^44 + w^2*x^42 + w^2*x^41 + w*x^39 +
w^2*x^38 + w*x^37 + w*x^36 + w^2*x^35 + w*x^33 + x^30 + w*x^29 + x^27 + w*x^26 + w*x^25 + x^23 + w*x^21 + w^2*x^19 + x^18 + w*x^17 + w^2*x^16 + w*x^13 +
w^2*x^12 + w*x^10 + x^9 + w*x^8 + w^2*x^7 + w^2*x^6 + w*x^5 + w*x^4 + w*x^3 + 1
[5]:  [85, 5, 61] Cyclic Linear Code over GF(2^2)
CyclicCode of length 85 with generating polynomial x^80 + x^78 + w^2*x^77 + w^2*x^76 + x^75 + w^2*x^74 + w^2*x^73 + w*x^72 + w^2*x^69 + x^67 + w*x^66 +
w*x^64 + w^2*x^63 + x^60 + w^2*x^59 + w*x^58 + w*x^56 + x^55 + w*x^54 + w*x^53 + w*x^52 + w^2*x^51 + x^50 + w^2*x^49 + w*x^48 + w^2*x^47 + x^46 + w*x^45 +
w^2*x^44 + w*x^43 + w^2*x^42 + w*x^41 + x^40 + w^2*x^38 + w^2*x^37 + w^2*x^36 + w*x^32 + w^2*x^31 + w*x^30 + w*x^29 + w^2*x^28 + w*x^26 + w^2*x^25 + x^24 + x^23
+ x^21 + w*x^20 + w*x^19 + x^17 + w^2*x^16 + w*x^14 + w*x^12 + w*x^11 + x^10 + x^9 + w^2*x^8 + w^2*x^7 + w*x^5 + w*x^4 + w*x + 1
[6]:  [85, 9, 53] Cyclic Linear Code over GF(2^2)
CyclicCode of length 85 with generating polynomial x^76 + w*x^75 + w^2*x^73 + w*x^72 + x^71 + w^2*x^69 + w*x^68 + w^2*x^67 + w*x^63 + x^62 + x^61 +
w^2*x^60 + x^59 + x^58 + w^2*x^57 + w*x^56 + w^2*x^55 + w^2*x^54 + w*x^53 + w^2*x^51 + x^50 + w^2*x^49 + w^2*x^48 + w^2*x^47 + w*x^46 + x^43 + x^42 + w*x^41 +
x^40 + x^39 + w^2*x^38 + w*x^36 + w^2*x^34 + w^2*x^33 + x^32 + x^31 + x^30 + w*x^28 + w*x^27 + w^2*x^26 + x^25 + w^2*x^24 + w^2*x^23 + w*x^22 + w*x^21 +
w^2*x^18 + w*x^17 + w^2*x^15 + w^2*x^14 + w^2*x^13 + x^12 + w*x^11 + w*x^10 + x^8 + w^2*x^7 + w^2*x^5 + x^4 + w^2*x^3 + x^2 + x + 1
[7]:  [98, 9, 62] Linear Code over GF(2^2)
ConstructionXX using [6] [5] [4] [3] and [1]
[8]:  [96, 9, 60] Linear Code over GF(2^2)
Puncturing of [7] at { 97 .. 98 }

```

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

```Lb(96,9) = 60 is found by truncation of:
Lb(97,9) = 61 MST

Ub(96,9) = 66 follows by a one-step Griesmer bound from:
Ub(29,8) = 16 follows by a one-step Griesmer bound from:
Ub(12,7) = 4 is found by construction B:
[consider deleting the (at most) 6 coordinates of a word in the dual]
```
###### References
MST: T. Maruta, M. Shinohara & M. Takenaka, Constructing linear codes from some orbits of projectivities, to appear in Discr. Math.

### 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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