Bounds on the minimum distance of linear codes

Bounds on linear codes [110,8] over GF(2)

Construction

Construction of a linear code [110,8,52] over GF(2):
[1]:  [4, 1, 4] Cyclic Linear Code over GF(2)
     RepetitionCode of length 4
[2]:  [4, 3, 2] Cyclic Linear Code over GF(2)
     Dual of the RepetitionCode of length 4
[3]:  [8, 4, 4] "Reed-Muller Code (r = 1, m = 3)" Linear Code over GF(2)
     PlotkinSum of [2] and [1]
[4]:  [7, 3, 4] Linear Code over GF(2)
     Shortening of [3] at 1
[5]:  [64, 4, 55] Linear Code over GF(2^3)
     BCHCode over GF(8) with parameters 63 54
[6]:  [448, 12, 220] Linear Code over GF(2)
     ConcatenatedCode of [5] and [4]
[7]:  [224, 11, 108] Linear Code over GF(2)
     generalized residue code of [6]
puncturing at the support of a word of weight 224
[8]:  [112, 10, 52] Linear Code over GF(2)
     generalized residue code of [7]
puncturing at the support of a word of weight 112
[9]:  [110, 8, 52] Linear Code over GF(2)
     Shortening of [8] at { 111 .. 112 }

last modified: 2001-01-30

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

Lb(110,8) = 52 is found by shortening of:
Lb(112,10) = 52 EB1

Ub(110,8) = 52 follows by a one-step Griesmer bound from:
Ub(57,7) = 26 otherwise adding a parity check bit would contradict:
Ub(58,7) = 27 vT3

References

vT3: H.C.A. van Tilborg, The smallest length of binary 7-dimensional linear codes with prescribed minimum distance, Discr. Math. 33 (1981) 197-207.

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