Bounds on the minimum distance of linear codes

Bounds on linear codes [171,5] over GF(3)

lower bound:113
upper bound:113

Construction

Construction of a linear code [171,5,113] over GF(3):
[1]: [39,0] Code
     ZeroCode of length 39
[2]:  [39, 1, 39] Cyclic Linear Code over GF(3)
     RepetitionCode of length 39
[3]:  [121, 116, 3] "Hamming code (r = 5)" Linear Code over GF(3)
      5-th order HammingCode over GF( 3)
[4]:  [121, 5, 81] Cyclic Linear Code over GF(3)
     Dual of [3]
[5]:  [119, 5, 79] Linear Code over GF(3)
     Puncturing of [4] at { 120 .. 121 }
[6]:  [40, 4, 27] Linear Code over GF(3)
     ResidueCode of [5]
[7]:  [39, 4, 26] Linear Code over GF(3)
     Puncturing of [6] at { 40 }
[8]:  [117, 5, 78] Linear Code over GF(3)
     PlotkinSum of [7] [2] and [1]
[9]:  [56, 6, 36] Linear Code over GF(3)
     QuasiTwistedCyclicCode of length 56 and constant 2 with generators: (1 0 0 0 0 0 2),  (2 1 1 0 1 0 0),  (1 1 0 2 0 2 1),  (1 1 1 0 0 1 0),  (1 2 0 1 0 2 
1),  (2 1 1 1 1 1 2),  (2 0 0 1 2 0 0),  (2 0 1 2 2 1 1)
[10]: [55, 5, 36] Linear Code over GF(3)
     Shortening of [9] at { 56 }
[11]: [172, 5, 114] Linear Code over GF(3)
     Juxtaposition of [10] and  [8]
[12]: [171, 5, 113] Linear Code over GF(3)
     Puncturing of [11] at { 172 }

last modified: 2001-12-17

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

Lb(171,5) = 113 is found by truncation of:
Lb(172,5) = 114 is found by concatenation of
[55,5,36] and [117,5,78]-codes

Ub(171,5) = 113 follows by the Griesmer bound.

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