Bounds on the minimum distance of linear codes

Bounds on linear codes [168,7] over GF(3)

lower bound:108
upper bound:108

Construction

Construction of a linear code [168,7,108] over GF(3):
[1]: [56,0] Code
     ZeroCode of length 56
[2]:  [56, 1, 56] Cyclic Linear Code over GF(3)
     RepetitionCode of length 56
[3]:  [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)
[4]:  [168, 7, 108] Linear Code over GF(3)
     PlotkinSum of [3] [2] and [1]

last modified: 2001-12-17

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

Lb(168,7) = 108 is found by the (u|u-v|u+v+w) construction
applied to [56,6,36] and [56,1,56] and empty code

Ub(168,7) = 108 follows by a one-step Griesmer bound from:
Ub(59,6) = 36 follows by a one-step Griesmer bound from:
Ub(22,5) = 12 is found by considering shortening to:
Ub(21,4) = 12 HN 
References
HN: R. Hill & D.E. Newton, Optimal ternary linear codes, Des. Codes Cryptogr. 2 (1992), 137-157.

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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