Bounds on the minimum distance of linear codes

Bounds on linear codes [213,8] over GF(3)

Construction

Construction of a linear code [213,8,135] over GF(3):
[1]:  [1, 1, 1] Cyclic Linear Code over GF(3)
     RepetitionCode of length 1
[2]:  [212, 204, 2] Linear Code over GF(3)
     Dual of [8]
[3]:  [212, 7, 135] Linear Code over GF(3)
     The intersection of [8] and [2]
[4]:  [2, 1, 2] Cyclic Linear Code over GF(3)
     RepetitionCode of length 2
[5]:  [208, 7, 132] Quasicyclic of degree 8 Linear Code over GF(3)
     QuasiCyclicCode of length 208 with generating polynomials: x^25 + 2*x^24 + 2*x^23 + 2*x^22 + x^21 + 2*x^20 + x^18 + 2*x^17 + 2*x^16 + x^15 + 2*x^11 + 2*x^10 + 2*x^8 + x,  x^23 + x^22 + 2*x^21 + x^19 + x^17 + x^15 + x^14 + 2*x^13 + 2*x^11 + 2*x^10 + 2*x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + 1,  x^25 + 2*x^24 + 2*x^23 + 2*x^22 + x^21 + x^20 + 2*x^19 + 2*x^17 + x^16 + x^12 + x^11 + 2*x^10 + 2*x^9 + 2*x^8 + 2*x^7 + x^4 + 2*x^3 + x^2 + 2*x + 1,  x^25 + x^23 + x^22 + x^21 + 2*x^17 + x^15 + x^12 + x^11 + 2*x^8 + 2*x^7 + 2*x^5 + 2*x^4 + 2*x^2 + x + 1,  x^25 + x^23 + x^21 + x^20 + x^19 + x^17 + 2*x^14 + 2*x^12 + x^11 + x^9 + x^8 + 2*x^7 + x^6 + 2*x^3 + 2*x^2 + 2*x + 1,  2*x^23 + x^21 + x^18 + x^17 + 2*x^14 + 2*x^13 + 2*x^11 + 2*x^10 + 2*x^8 + x^7 + x^6 + x^5 + x^3 + x^2 + x,  x^24 + x^23 + 2*x^21 + 2*x^20 + x^18 + 2*x^17 + x^16 + x^14 + x^12 + 2*x^11 + x^9 + 2*x^8 + x^5 + x^4 + 2*x^2 + x + 1,  2*x^24 + x^22 + 2*x^21 + x^20 + 2*x^19 + 2*x^17 + x^16 + x^15 + 2*x^14 + x^13 + 2*x^12 + x^11 + 2*x^10 + x^9 + x^6 + x^5 + x^4 + x^3 + 2*x + 1
[6]:  [208, 7, 132] Quasicyclic of degree 8 Linear Code over GF(3)
     QuasiCyclicCode of length 208 with generating polynomials: 2*x^25 + 2*x^23 + x^22 + x^21 + x^20 + 2*x^19 + x^18 + 2*x^16 + x^15 + x^14 + 2*x^13 + x^9 + x^8 + x^6,  2*x^24 + 2*x^21 + 2*x^20 + x^19 + 2*x^17 + 2*x^15 + 2*x^13 + 2*x^12 + x^11 + x^9 + x^8 + x^7 + 2*x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2,  2*x^25 + 2*x^22 + 2*x^21 + 2*x^20 + 2*x^17 + x^16 + 2*x^15 + x^13 + x^12 + x^11 + 2*x^8 + 2*x^7 + 2*x^6 + 2*x^5 + x^4 + x^3 + 2*x,  x^25 + x^24 + x^23 + 2*x^22 + x^21 + x^20 + x^19 + 2*x^18 + 2*x^17 + 2*x^16 + 2*x^14 + x^13 + 2*x^12 + 2*x^11 + x^10 + x^9 + 2*x^8 + 2*x^7 + 2*x^4 + 2*x,  2*x^25 + x^22 + x^20 + x^16 + x^14 + x^13 + 2*x^12 + x^11 + 2*x^10 + x^8 + 2*x^5 + x^3 + x^2 + 2*x + 2,  x^24 + x^23 + x^22 + 2*x^21 + x^20 + x^18 + x^17 + x^14 + x^13 + 2*x^12 + 2*x^11 + x^10 + 2*x^9 + 2*x^8 + x^7 + 2*x^6 + x^2,  2*x^25 + 2*x^24 + 2*x^22 + 2*x^21 + x^19 + x^18 + 2*x^16 + x^15 + 2*x^14 + 2*x^12 + 2*x^10 + x^9 + 2*x^7 + x^6 + 2*x^3 + 2*x^2 + 1,  x^24 + 2*x^23 + 2*x^21 + x^20 + x^18 + 2*x^16 + x^14 + x^13 + x^11 + x^9 + x^7 + 2*x^6 + 2*x^5 + x^4 + x^3 + x^2 + x + 2
[7]:  [208, 8, 130] Quasicyclic of degree 8 Linear Code over GF(3)
     The Vector space sum: [6] + [5]
[8]:  [212, 8, 134] Linear Code over GF(3)
     ConstructionXX using [7] [6] [5] [4] and [4]
[9]:  [213, 8, 135] Linear Code over GF(3)
     ConstructionX using [8] [3] and [1]

last modified: 2008-11-06

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

Lb(213,8) = 132 is found by truncation of:
Lb(216,8) = 135 BZ 

Ub(213,8) = 138 follows by a one-step Griesmer bound from:
Ub(74,7) = 45 follows by a one-step Griesmer bound from:
Ub(28,6) = 15 HHM

References

HHM: N. Hamada, T. Helleseth, H.M. Martinsen & Ø. Ytrehus, There is no ternary [28,6,16] code

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