Bounds on the minimum distance of linear codes

Bounds on linear codes [212,14] over GF(2)

Construction

Construction of a linear code [212,14,96] over GF(2):
[1]:  [23, 12, 7] "Quadratic Residue code" Linear Code over GF(2)
     QRCode over GF(2)of length 23
[2]:  [23, 11, 8] Cyclic Linear Code over GF(2)
     Dual of [1]
[3]:  [19, 7, 8] Linear Code over GF(2)
     Shortening of [2] at { 20, 21, 22, 23 }
[4]:  [63, 4, 47] "BCH code (d = 46, b = 1)" Linear Code over GF(2^2)
     BCHCode over GF(4) with parameters 63 46
[5]:  [189, 8, 94] Quasicyclic of degree 63 Linear Code over GF(2)
     ConcatenatedCode of [4] and [16]
[6]:  [192, 8, 96] Quasicyclic of degree 64 Linear Code over GF(2)
     The Generator Elements of [5] with respect to the basis of [15]
encoded to new generator elements using [13]
[7]:  [194, 8, 96] Linear Code over GF(2)
     The Generator Elements of [6] with respect to the basis of [20]
encoded to new generator elements using [21]
[8]:  [2, 1, 2] Cyclic Linear Code over GF(2)
     RepetitionCode of length 2
[9]:  [192, 2, 102] Quasicyclic of degree 64 Linear Code over GF(2)
     CodeComplement of [13] with [19]
[10]: [192, 1, 102] Linear Code over GF(2)
     Subcode of [9]
[11]: [63, 7, 43] "BCH code (d = 42, b = 63)" Linear Code over GF(2^2)
     BCHCode over GF(4) with parameters 63 42 0
[12]: [189, 14, 86] Quasicyclic of degree 63 Linear Code over GF(2)
     ConcatenatedCode of [11] and [16]
[13]: [192, 16, 86] Quasicyclic of degree 64 Linear Code over GF(2)
     ConstructionX using [15] [12] and [16]
[14]: [63, 8, 42] "BCH code (d = 42, b = 1)" Linear Code over GF(2^2)
     BCHCode over GF(4) with parameters 63 42
[15]: [189, 16, 84] Quasicyclic of degree 63 Linear Code over GF(2)
     ConcatenatedCode of [14] and [16]
[16]: [3, 2, 2] Cyclic Linear Code over GF(2)
     CordaroWagnerCode of length 3
[17]: [63, 7, 43] "BCH code (d = 43, b = 1)" Linear Code over GF(2^2)
     BCHCode over GF(4) with parameters 63 43
[18]: [189, 14, 86] Quasicyclic of degree 63 Linear Code over GF(2)
     ConcatenatedCode of [17] and [16]
[19]: [192, 14, 88] Quasicyclic of degree 64 Linear Code over GF(2)
     The Generator Elements of [18] with respect to the basis of [15]
encoded to new generator elements using [13]
[20]: [192, 15, 86] Linear Code over GF(2)
     The Vector space sum: [19] + [10]
[21]: [194, 15, 88] Linear Code over GF(2)
     ConstructionX using [20] [19] and [8]
[22]: [213, 15, 96] Linear Code over GF(2)
     ConstructionX using [21] [7] and [3]
[23]: [212, 14, 96] Linear Code over GF(2)
     Shortening of [22] at { 213 }

last modified: 2001-01-30

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

Lb(212,14) = 96 is found by shortening of:
Lb(213,15) = 96 EB1

Ub(212,14) = 99 is found by considering shortening to:
Ub(211,13) = 99 BK 

References

EB1: Y. Edel & J. Bierbrauer, Some codes related to BCH codes of low dimension, preprint, 1995.

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