Bounds on the minimum distance of linear codes

Bounds on linear codes [130,5] over GF(5)

Construction

Construction of a linear code [130,5,100] over GF(5):
[1]:  [156, 152, 3] "Hamming code (r = 4)" Linear Code over GF(5)
      4-th order HammingCode over GF( 5)
[2]:  [156, 4, 125] Linear Code over GF(5)
     Dual of [1]
[3]:  [152, 4, 121] Linear Code over GF(5)
     Puncturing of [2] at { 153 .. 156 }
[4]:  [31, 3, 25] Linear Code over GF(5)
     ResidueCode of [3]
[5]:  [27, 3, 21] Linear Code over GF(5)
     Puncturing of [4] at { 28 .. 31 }
[6]:  [6, 2, 5] Linear Code over GF(5)
     ResidueCode of [5]
[7]:  [124, 3, 100] Cyclic Linear Code over GF(5)
     CyclicCode of length 124 with generating polynomial x^121 + 3*x^120 + x^118 + 2*x^117 + 2*x^116 + x^115 + x^114 + 4*x^112 + 4*x^110 + 4*x^109 + x^108 + 
4*x^107 + 2*x^105 + 3*x^104 + x^103 + 2*x^102 + x^101 + 3*x^100 + x^99 + 4*x^98 + 2*x^97 + 3*x^96 + 3*x^95 + 3*x^94 + x^93 + 3*x^90 + 4*x^89 + 3*x^87 + x^86 + 
x^85 + 3*x^84 + 3*x^83 + 2*x^81 + 2*x^79 + 2*x^78 + 3*x^77 + 2*x^76 + x^74 + 4*x^73 + 3*x^72 + x^71 + 3*x^70 + 4*x^69 + 3*x^68 + 2*x^67 + x^66 + 4*x^65 + 4*x^64
+ 4*x^63 + 3*x^62 + 4*x^59 + 2*x^58 + 4*x^56 + 3*x^55 + 3*x^54 + 4*x^53 + 4*x^52 + x^50 + x^48 + x^47 + 4*x^46 + x^45 + 3*x^43 + 2*x^42 + 4*x^41 + 3*x^40 + 
4*x^39 + 2*x^38 + 4*x^37 + x^36 + 3*x^35 + 2*x^34 + 2*x^33 + 2*x^32 + 4*x^31 + 2*x^28 + x^27 + 2*x^25 + 4*x^24 + 4*x^23 + 2*x^22 + 2*x^21 + 3*x^19 + 3*x^17 + 
3*x^16 + 2*x^15 + 3*x^14 + 4*x^12 + x^11 + 2*x^10 + 4*x^9 + 2*x^8 + x^7 + 2*x^6 + 3*x^5 + 4*x^4 + x^3 + x^2 + x + 2
[8]:  [124, 6, 95] Cyclic Linear Code over GF(5)
     CyclicCode of length 124 with generating polynomial x^118 + 4*x^117 + x^116 + 4*x^115 + 4*x^114 + 3*x^113 + 3*x^112 + x^111 + 4*x^110 + 2*x^109 + 4*x^108 +
3*x^107 + 3*x^106 + x^105 + 3*x^104 + 2*x^103 + 4*x^102 + 3*x^101 + 4*x^99 + 2*x^98 + 3*x^97 + 4*x^96 + 2*x^95 + 4*x^94 + 2*x^93 + x^92 + 4*x^91 + 2*x^89 + 
3*x^88 + 3*x^86 + 4*x^84 + 2*x^83 + x^82 + 4*x^81 + 2*x^80 + 4*x^79 + x^78 + 2*x^77 + 2*x^76 + 2*x^75 + 2*x^74 + x^73 + 3*x^72 + 4*x^71 + 3*x^70 + x^69 + 4*x^68
+ x^67 + 2*x^66 + 3*x^65 + 3*x^64 + 3*x^63 + 2*x^61 + 3*x^60 + 4*x^58 + x^57 + 3*x^56 + 3*x^55 + 3*x^54 + x^52 + x^51 + 2*x^50 + 2*x^49 + 3*x^47 + x^46 + 3*x^45
+ 3*x^44 + 2*x^43 + x^42 + 2*x^41 + 2*x^38 + 3*x^36 + 3*x^35 + 3*x^34 + 2*x^33 + 3*x^32 + x^31 + 2*x^30 + 3*x^29 + 4*x^27 + x^26 + x^25 + x^23 + 2*x^22 + 3*x^21
+ x^19 + 2*x^17 + 4*x^16 + 3*x^15 + 2*x^14 + 2*x^13 + 3*x^10 + 2*x^9 + 4*x^8 + 2*x^7 + 2*x^6 + 4*x^5 + 2*x^4 + 3*x^2 + 2
[9]:  [124, 5, 95] Linear Code over GF(5)
     SubcodeBetweenCode of dimension 5 of [8] and [7]
[10]: [130, 5, 100] Linear Code over GF(5)
     ConstructionX using [9] [7] and [6]

last modified: 2003-07-02

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

Lb(130,5) = 100 is found by lengthening of:
Lb(129,5) = 100 Koh

Ub(130,5) = 101 follows by a one-step Griesmer bound from:
Ub(28,4) = 20 follows by a one-step Griesmer bound from:
Ub(7,3) = 4 is found by construction B:
[consider deleting the (at most) 3 coordinates of a word in the dual]

References

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