Bounds on the minimum distance of linear codes

Bounds on linear codes [249,21] over GF(2)

lower bound:106
upper bound:112


Construction type: GraX

Construction of a linear code [249,21,106] over GF(2):
[1]:  [9, 8, 2] Cyclic Linear Code over GF(2)
     Dual of the RepetitionCode of length 9
[2]:  [240, 13, 112] "Goppa code (r = 102)" Linear Code over GF(2)
     GoppaCode with 240 points over GF(256) and polynomial x^102 + x^72 + x^42 + x^12 where w := GF(256).1
[3]:  [240, 21, 104] "Goppa code (r = 96)" Linear Code over GF(2)
     GoppaCode with 240 points over GF(256) and polynomial x^96 + x^66 + x^36 + x^6 where w := GF(256).1
[4]:  [249, 21, 106] Linear Code over GF(2)
     ConstructionX using [3] [2] and [1]

last modified: 2020-10-18

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

Lb(249,21) = 105 is found by truncation of:
Lb(256,21) = 112 XBC

Ub(249,21) = 113 follows by a one-step Griesmer bound from:
Ub(135,20) = 56 is found by considering shortening to:
Ub(132,17) = 56 otherwise adding a parity check bit would contradict:
Ub(133,17) = 57 LP 
LP: Follows from the linear programming bound.

XBC: Extended BCH code.


  • 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]

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This page is maintained by Markus Grassl ( Last change: 30.12.2011