Bounds on the minimum distance of linear codes

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

Construction

Construction of a linear code [196,17,82] over GF(2):
[1]:  [9, 1, 9] Cyclic Linear Code over GF(2)
     RepetitionCode of length 9
[2]:  [195, 16, 88] Quasicyclic of degree 13 Linear Code over GF(2)
     QuasiCyclicCode of length 195 stacked to height 3 with generating polynomials: x^13 + x^12 + x^11 + x^10 + x^6 + x^3 + x^2 + 1,  x^12 + x^10 + x^6 + x^5 + x^2 + x,  x^9 + x^6 + x^4 + x,  x^10 + x^6 + x + 1,  x^12 + x^10 + x^8 + x^6 + x^3 + x^2 + x + 1,  x^13 + x^11 + x^9 + x^8 + x^6 + x^4,  x^13 + x^12 + x^11 + x^9 + x^8 + x^6 + x^5 + x^4 + x^2 + 1,  x^7 + x^6 + x^5 + x^2 + x + 1,  x^12 + x^9 + x^7 + x^4,  x^13 + x^12 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x + 1,  x^13 + x^11 + x^9 + x^6 + x^5 + x^4 + x^3 + 1,  x^11 + x^10 + x^7 + x^6 + x^2 + 1,  x^12 + x^9 + x^8 + x^6 + x^4 + x^3 + x^2 + x,  x^13 + x^10 + x^9 + x^7 + x^5 + x^4 + x^3 + x^2,  x^13 + x^12 + x^11 + x^10 + x^6 + x^3 + x^2 + 1,  x^12 + x^10 + x^6 + x^5 + x^2 + x,  x^9 + x^6 + x^4 + x,  x^10 + x^6 + x + 1,  x^12 + x^10 + x^8 + x^6 + x^3 + x^2 + x + 1,  x^13 + x^11 + x^9 + x^8 + x^6 + x^4,  x^13 + x^12 + x^11 + x^9 + x^8 + x^6 + x^5 + x^4 + x^2 + 1,  x^7 + x^6 + x^5 + x^2 + x + 1,  x^12 + x^9 + x^7 + x^4,  x^13 + x^12 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x + 1,  x^13 + x^11 + x^9 + x^6 + x^5 + x^4 + x^3 + 1,  x^11 + x^10 + x^7 + x^6 + x^2 + 1,  x^12 + x^11 + x^8 + x^7 + x^3 + x,  x^13 + x^10 + x^9 + x^7 + x^5 + x^4 + x^3 + x^2,  x^13 + x^12 + x^11 + x^10 + x^6 + x^3 + x^2 + 1,  x^12 + x^10 + x^6 + x^5 + x^2 + x,  x^9 + x^6 + x^4 + x,  x^10 + x^6 + x + 1,  x^12 + x^10 + x^8 + x^6 + x^3 + x^2 + x + 1,  x^13 + x^11 + x^9 + x^8 + x^6 + x^4,  x^13 + x^12 + x^11 + x^9 + x^8 + x^6 + x^5 + x^4 + x^2 + 1,  x^7 + x^6 + x^5 + x^2 + x + 1,  x^12 + x^9 + x^7 + x^4,  x^13 + x^12 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x + 1,  x^13 + x^11 + x^9 + x^6 + x^5 + x^4 + x^3 + 1
[3]:  [195, 17, 79] Quasicyclic of degree 13 Linear Code over GF(2)
     QuasiCyclicCode of length 195 stacked to height 4 with generating polynomials: x^13 + x^12 + x^11 + x^10 + x^6 + x^3 + x^2 + 1,  x^12 + x^10 + x^6 + x^5 + x^2 + x,  x^9 + x^6 + x^4 + x,  x^10 + x^6 + x + 1,  x^12 + x^10 + x^8 + x^6 + x^3 + x^2 + x + 1,  x^13 + x^11 + x^9 + x^8 + x^6 + x^4,  x^13 + x^12 + x^11 + x^9 + x^8 + x^6 + x^5 + x^4 + x^2 + 1,  x^7 + x^6 + x^5 + x^2 + x + 1,  x^12 + x^9 + x^7 + x^4,  x^13 + x^12 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x + 1,  x^13 + x^11 + x^9 + x^6 + x^5 + x^4 + x^3 + 1,  x^11 + x^10 + x^7 + x^6 + x^2 + 1,  x^12 + x^9 + x^8 + x^6 + x^4 + x^3 + x^2 + x,  x^13 + x^10 + x^9 + x^7 + x^5 + x^4 + x^3 + x^2,  x^13 + x^12 + x^11 + x^10 + x^6 + x^3 + x^2 + 1,  x^12 + x^10 + x^6 + x^5 + x^2 + x,  x^9 + x^6 + x^4 + x,  x^10 + x^6 + x + 1,  x^12 + x^10 + x^8 + x^6 + x^3 + x^2 + x + 1,  x^13 + x^11 + x^9 + x^8 + x^6 + x^4,  x^13 + x^12 + x^11 + x^9 + x^8 + x^6 + x^5 + x^4 + x^2 + 1,  x^7 + x^6 + x^5 + x^2 + x + 1,  x^12 + x^9 + x^7 + x^4,  x^13 + x^12 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x + 1,  x^13 + x^11 + x^9 + x^6 + x^5 + x^4 + x^3 + 1,  x^11 + x^10 + x^7 + x^6 + x^2 + 1,  x^12 + x^11 + x^8 + x^7 + x^3 + x,  x^13 + x^10 + x^9 + x^7 + x^5 + x^4 + x^3 + x^2,  x^13 + x^12 + x^11 + x^10 + x^6 + x^3 + x^2 + 1,  x^12 + x^10 + x^6 + x^5 + x^2 + x,  x^9 + x^6 + x^4 + x,  x^10 + x^6 + x + 1,  x^12 + x^10 + x^8 + x^6 + x^3 + x^2 + x + 1,  x^13 + x^11 + x^9 + x^8 + x^6 + x^4,  x^13 + x^12 + x^11 + x^9 + x^8 + x^6 + x^5 + x^4 + x^2 + 1,  x^7 + x^6 + x^5 + x^2 + x + 1,  x^12 + x^9 + x^7 + x^4,  x^13 + x^12 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x + 1,  x^13 + x^11 + x^9 + x^6 + x^5 + x^4 + x^3 + 1,  x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,  x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,  x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,  0,  x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,  0,  x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,  x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,  x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,  x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,  0,  0,  x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
[4]:  [204, 17, 88] Linear Code over GF(2)
     ConstructionX using [3] [2] and [1]
[5]:  [195, 17, 82] Linear Code over GF(2)
     Puncturing of [4] at { 46, 47, 48, 51, 52, 53, 56, 57, 58 }
[6]:  [196, 17, 82] Linear Code over GF(2)
     ExtendCode [5] by 1

last modified: 2015-03-10

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

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

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