Bounds on the minimum distance of linear codes

Bounds on linear codes [196,17] over GF(2)

lower bound:81
upper bound:88

Construction

Construction of a linear code 
[196,17,81] over GF(2):
[1]:  [10, 1, 10] Cyclic Linear Code over GF(2)
     RepetitionCode of length 10
[2]:  [195, 16, 88] Cyclic Linear Code over GF(2)
     CyclicCode of length 195 with generating polynomial x^179 + x^178 + x^175 +
x^174 + x^169 + x^168 + x^165 + x^164 + x^162 + x^160 + x^157 + x^156 + x^154 + 
x^153 + x^149 + x^148 + x^143 + x^141 + x^134 + x^133 + x^131 + x^130 + x^129 + 
x^127 + x^126 + x^125 + x^123 + x^122 + x^119 + x^116 + x^113 + x^110 + x^109 + 
x^108 + x^102 + x^100 + x^99 + x^98 + x^90 + x^89 + x^88 + x^87 + x^85 + x^84 + 
x^83 + x^82 + x^81 + x^80 + x^79 + x^78 + x^75 + x^74 + x^72 + x^71 + x^66 + 
x^64 + x^62 + x^61 + x^60 + x^58 + x^57 + x^54 + x^51 + x^49 + x^43 + x^39 + 
x^38 + x^37 + x^33 + x^32 + x^30 + x^27 + x^26 + x^25 + x^22 + x^20 + x^17 + 
x^16 + x^15 + x^14 + x^11 + x^10 + x^9 + x^7 + x^6 + x^4 + x^3 + 1
[3]:  [195, 18, 78] Cyclic Linear Code over GF(2)
     CyclicCode of length 195 with generating polynomial x^177 + x^175 + x^174 +
x^173 + x^172 + x^170 + x^169 + x^166 + x^165 + x^162 + x^161 + x^160 + x^155 + 
x^153 + x^147 + x^145 + x^144 + x^142 + x^140 + x^138 + x^137 + x^135 + x^134 + 
x^131 + x^130 + x^129 + x^128 + x^127 + x^124 + x^122 + x^117 + x^116 + x^111 + 
x^110 + x^107 + x^105 + x^104 + x^102 + x^101 + x^100 + x^97 + x^95 + x^94 + 
x^92 + x^91 + x^89 + x^86 + x^84 + x^81 + x^78 + x^73 + x^71 + x^64 + x^63 + 
x^62 + x^59 + x^57 + x^52 + x^51 + x^47 + x^46 + x^44 + x^43 + x^37 + x^31 + 
x^29 + x^27 + x^26 + x^25 + x^24 + x^23 + x^21 + x^19 + x^17 + x^16 + x^15 + 
x^14 + x^13 + x^12 + x^10 + x^7 + x^6 + x^5 + x^4 + x + 1
[4]:  [195, 17, 78] Linear Code over GF(2)
     SubcodeBetweenCode of dimension 17 of [3] and [2]
[5]:  [205, 17, 88] Linear Code over GF(2)
     ConstructionX using [4] [2] and [1]
[6]:  [199, 17, 84] Linear Code over GF(2)
     Puncturing of [5] at { 1, 40, 79, 118, 157, 196 }
[7]:  [196, 17, 81] Linear Code over GF(2)
     Puncturing of [6] at { 197 .. 199 }

last modified: 2004-08-20

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

Lb(196,17) = 80 is found by taking a subcode of:
Lb(196,18) = 80 is found by lengthening of:
Lb(193,18) = 80 GW2

Ub(196,17) = 88 otherwise adding a parity check bit would contradict:
Ub(197,17) = 89 BK 
References
BK: Detlef Berntzen & Peter Kemper, email, Feb. 1993.

GW2: M. Grassl & G. White, New Codes from Chains of Quasi-cyclic Codes, ISIT 2005.

Notes

  • All codes establishing the lower bounds where constructed using MAGMA.
  • Upper bounds are taking 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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