Bounds on the minimum distance of linear codes
Bounds on linear codes [68,11] over GF(5)
lower bound:  40 
upper bound:  46 
Construction
Construction type: EricChen
Construction of a linear code [68,11,40] over GF(5):
[1]: [2, 1, 2] Cyclic Linear Code over GF(5)
RepetitionCode of length 2
[2]: [66, 10, 40] Quasicyclic of degree 3 Linear Code over GF(5)
QuasiCyclicCode of length 66 with generating polynomials: x^21 + 2*x^20 + 4*x^19 + x^18 + 4*x^17 + 2*x^15 + 4*x^13 + 4*x^12 + 4*x^11 + 3*x^10 + x^9, 4*x^21 + 2*x^20 + 3*x^19 + 3*x^18 + 3*x^17 + x^16 + 4*x^15 + x^14 + 3*x^13 + 2*x^12 + 3*x^11 + 4*x^10 + 3*x^9 + x^8 + 3*x^7 + 4*x^6 + x^4 + x^3 + x^2 + 3*x, 4*x^21 + 2*x^20 + 3*x^19 + 4*x^18 + 4*x^17 + 2*x^15 + x^13 + 4*x^12 + x^11 + x^10 + 4*x^9 + x^8 + x^7 + x^6 + x^2 + 1
[3]: [66, 11, 38] Quasicyclic of degree 3 Linear Code over GF(5)
QuasiCyclicCode of length 66 with generating polynomials: x^21 + 2*x^20 + x^19 + 4*x^18 + 3*x^17 + x^16 + 2*x^15 + 3*x^14 + x^13 + x^12 + 2*x^11 + x^5, x^21 + 3*x^20 + 3*x^19 + 2*x^18 + 3*x^17 + 2*x^16 + 2*x^15 + x^14 + 2*x^13 + 2*x^12 + x^11 + 2*x^10 + 2*x^9 + 4*x^8 + 2*x^6 + 4*x^5 + 4*x^3 + 4*x^2 + x + 1, 2*x^21 + x^20 + x^19 + 3*x^18 + 3*x^17 + 2*x^16 + 4*x^14 + x^13 + 4*x^12 + x^11 + 4*x^10 + x^9 + 2*x^8 + 2*x^7 + 3*x^5 + 2*x^4 + 2*x^3 + x^2 + 4*x + 2
[4]: [68, 11, 40] Linear Code over GF(5)
ConstructionX using [3] [2] and [1]
last modified: 20210825
From Brouwer's table (as of 20070213)
Lb(68,11) = 39 is found by truncation of:
Lb(69,11) = 40 Ma
Ub(68,11) = 46 is found by considering shortening to:
Ub(66,9) = 46 is found by considering truncation to:
Ub(65,9) = 45 is found by construction B:
[consider deleting the (at most) 6 coordinates of a word in the dual]
References
Ma:
T. Maruta, On the nonexistence of linear codes attaining the Griesmer
bound, Geom. Dedicata 60 (1996) 17.
T. Maruta, On the nonexistence of linear codes of dimension four attaining
the Griesmer bound, pp. 117120 in: Optimal codes and related topics, Proc.
Workshop Sozopol, Bulgaria, 1995.
T. Maruta, The nonexistence of [116,5,85]_4 codes and [187,5,139]_4
codes, Proc. 2nd International Workshop on Optimal Codes and Related Topics
in Sozopol (1998), pp. 168174.
T. Maruta & M. Fukui, On the nonexistence of some linear codes of dimension 4
over GF(5), preprint, 1995.
T. Maruta, M. Takenaka, M. Shinohara, K. Masuda & S. Kawashima, Constructing
new linear codes over small fields, preprint 2004.
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 20012006.
 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 email:
codes [at] codetables.de

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Last change: 30.12.2011