Bounds on the minimum distance of linear codes

Bounds on linear codes [213,6] over GF(3)

Construction

Construction of a linear code [213,6,139] over GF(3):
[1]: [9,0] Code
     ZeroCode of length 9
[2]:  [9, 1, 9] Cyclic Linear Code over GF(3)
     RepetitionCode of length 9
[3]:  [12, 6, 6] Linear Code over GF(3)
     Extend the QRCode over GF(3)of length 11
[4]:  [9, 3, 6] Linear Code over GF(3)
     Shortening of [3] at { 10 .. 12 }
[5]:  [27, 4, 18] Linear Code over GF(3)
     PlotkinSum of [4] [2] and [1]
[6]:  [192, 2, 144] Quasicyclic of degree 24 Linear Code over GF(3)
     QuasiCyclicCode of length 192 with generating polynomials: x^7 + 2*x^6 + 2*x^4 + 2*x^3 + x^2 + 1,  2*x^6 + 2*x^5 + x^4 + x^2 + x + 2,  x^7 + 2*x^6 + 2*x^4 + 2*x^3 + x^2 + 1,  x^7 + 2*x^6 + 2*x^4 + 2*x^3 + x^2 + 1,  x^6 + x^5 + 2*x^4 + 2*x^2 + 2*x + 1,  2*x^7 + 2*x^5 + 2*x^4 + x^3 + x + 1,  x^6 + x^5 + 2*x^4 + 2*x^2 + 2*x + 1,  x^7 + x^6 + 2*x^5 + 2*x^3 + 2*x^2 + x,  2*x^7 + 2*x^6 + x^5 + x^3 + x^2 + 2*x,  x^6 + x^5 + 2*x^4 + 2*x^2 + 2*x + 1,  x^7 + x^5 + x^4 + 2*x^3 + 2*x + 2,  x^6 + x^5 + 2*x^4 + 2*x^2 + 2*x + 1,  x^6 + x^5 + 2*x^4 + 2*x^2 + 2*x + 1,  x^7 + 2*x^6 + 2*x^4 + 2*x^3 + x^2 + 1,  2*x^7 + 2*x^6 + x^5 + x^3 + x^2 + 2*x,  2*x^7 + x^6 + x^4 + x^3 + 2*x^2 + 2,  x^7 + x^6 + 2*x^5 + 2*x^3 + 2*x^2 + x,  2*x^6 + 2*x^5 + x^4 + x^2 + x + 2,  2*x^7 + 2*x^5 + 2*x^4 + x^3 + x + 1,  2*x^7 + x^6 + x^4 + x^3 + 2*x^2 + 2,  x^7 + 2*x^6 + 2*x^4 + 2*x^3 + x^2 + 1,  x^7 + x^5 + x^4 + 2*x^3 + 2*x + 2,  x^7 + x^6 + 2*x^5 + 2*x^3 + 2*x^2 + x,  x^7 + x^5 + x^4 + 2*x^3 + 2*x + 2
[7]:  [192, 6, 126] Quasicyclic of degree 24 Linear Code over GF(3)
     QuasiCyclicCode of length 192 with generating polynomials: x^7 + 2*x^6 + 1,  2*x^7 + x^6 + 2*x^5 + 2*x^3 + 2*x^2 + 2,  x^7 + x^6 + 2*x^5 + 2*x^2 + 2*x + 1,  x^7 + 2*x^6 + 2*x^5 + x^4 + 2*x + 2,  x^7 + 2*x^5 + x^3 + 2*x^2 + x + 1,  2*x^7 + x^5 + x^4 + 2*x + 2,  x^7 + x^6 + x^5 + x^3 + 1,  2*x^6 + x^5 + 2*x^3 + x^2 + 2*x,  2*x^6 + x^3 + 2*x,  x^7 + x^6 + 2*x^5 + x^3 + x + 1,  2*x^7 + x^6 + x^5 + 2*x^4 + x^3 + x^2 + 1,  x^7 + 2*x^5 + 2*x^4 + x^3 + 2*x^2 + x,  x^7 + x^5 + 2*x^4 + x^3 + 2*x^2,  x^7 + x^6 + 2*x^2 + 1,  2*x^6 + 2*x^5 + x^4 + x^3 + x + 1,  2*x^7 + 2*x^2 + 2,  2*x^6 + 2*x^4 + 2*x^3 + x^2 + x + 2,  2*x^7 + 2*x^3 + x^2 + x + 2,  x^7 + 2*x^6 + x^5 + x^4 + 2*x^3 + 2*x^2 + 2*x + 2,  2*x^7 + 2*x^4 + 2*x^2 + 1,  x^7 + 2*x^6 + x^4 + 2,  2*x^7 + x^6 + x^5 + x^4 + x^3 + x^2,  x^6 + x^5 + 2*x^3 + 2*x,  2*x^7 + x^6 + 2*x^4 + x^3 + x^2 + 2*x + 1
[8]:  [219, 6, 144] Linear Code over GF(3)
     ConstructionX using [7] [6] and [5]
[9]:  [215, 6, 141] Linear Code over GF(3)
     Puncturing of [8] at { 5, 109, 125, 201 }
[10]: [213, 6, 139] Linear Code over GF(3)
     Puncturing of [9] at { 214 .. 215 }

last modified: 2004-07-27

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

Lb(213,6) = 139 is found by truncation of:
Lb(215,6) = 141 Koh

Ub(213,6) = 140 follows by a one-step Griesmer bound from:
Ub(72,5) = 46 is found by considering truncation to:
Ub(71,5) = 45 HW1

References

Koh: Axel Kohnert, email, 2006.

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