lower bound: | 69 |
upper bound: | 75 |
Construction of a linear code [126,12,69] over GF(3): [1]: [121, 116, 3] "Hamming code (r = 5)" Linear Code over GF(3) 5-th order HammingCode over GF( 3) [2]: [121, 5, 81] Cyclic Linear Code over GF(3) Dual of [1] [3]: [119, 5, 79] Linear Code over GF(3) Puncturing of [2] at { 120 .. 121 } [4]: [40, 4, 27] Linear Code over GF(3) ResidueCode of [3] [5]: [38, 4, 25] Linear Code over GF(3) Puncturing of [4] at { 39 .. 40 } [6]: [13, 3, 9] Linear Code over GF(3) ResidueCode of [5] [7]: [11, 3, 7] Linear Code over GF(3) Puncturing of [6] at { 12 .. 13 } [8]: [4, 2, 3] Linear Code over GF(3) ResidueCode of [7] [9]: [1, 1, 1] Cyclic Linear Code over GF(3) RepetitionCode of length 1 [10]: [121, 10, 69] Quasicyclic of degree 11 Linear Code over GF(3) QuasiCyclicCode of length 121 with generating polynomials: 2*x^10 + x^3, x^10 + 2*x^9 + 2*x^6 + 2*x^5 + 2*x^2 + x + 2, 2*x^10 + x^9 + 2*x^8 + x^7 + x^5 + 2*x^4 + x^3 + x^2 + 2*x + 2, 2*x^8 + x^7 + x^6 + x^5 + 2*x^4 + 2*x^2 + 2*x + 1, x^10 + x^9 + 2*x^8 + x^7 + 2*x^6 + x^5 + x^2 + x + 2, 2*x^10 + 2*x^9 + 2*x^8 + x^6 + x^3 + 1, x^10 + 2*x^6 + 2*x^5 + x^3 + x^2 + 2, 2*x^9 + 2*x^6 + x^3 + x^2 + 2*x + 1, 2*x^10 + x^9 + x^8 + x^6 + x^4 + x^2 + 2*x, x^8 + x^7 + x^6 + x^5 + 2*x^4 + 2*x^2 + 2*x + 2, 2*x^10 + x^8 + x^5 + x^4 + 2*x^3 + 2*x [11]: [121, 11, 66] Quasicyclic of degree 11 Linear Code over GF(3) QuasiCyclicCode of length 121 with generating polynomials: x^2, x^9 + x^8 + x^7 + x^3 + 2*x + 1, x^10 + 2*x^9 + x^8 + 2*x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 1, x^10 + 2*x^9 + 2*x^8 + x^7 + 2*x^6 + 2*x^2 + 2*x + 2, x^10 + x^8 + 2*x^7 + 2*x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2, 2*x^7 + x^5 + 2*x^3 + x^2 + x, 2*x^9 + 2*x^8 + x^7 + x^5 + x^4 + x^3 + x^2 + 2*x + 1, x^9 + 2*x^7 + 2*x^3 + x^2 + x + 2, x^10 + x^6 + x^5 + x^3 + x^2 + 2*x + 2, x^10 + 2*x^9 + 2*x^8 + 2*x^6 + 2*x^3 + 2*x^2 + 2*x + 2, 2*x^10 + x^7 + 2*x^6 + x^4 + 2*x^3 + x^2 [12]: [121, 12, 65] Linear Code over GF(3) QuasiCyclicCode of length 121 stacked to height 2 with generating polynomials: 0, 0, x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, 2*x^10 + 2*x^9 + 2*x^8 + 2*x^7 + 2*x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 2*x + 2, x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, 2*x^10 + 2*x^9 + 2*x^8 + 2*x^7 + 2*x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 2*x + 2, 2*x^10 + 2*x^9 + 2*x^8 + 2*x^7 + 2*x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 2*x + 2, x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, 0, 0, x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, x^2, x^9 + x^8 + x^7 + x^3 + 2*x + 1, x^10 + 2*x^9 + x^8 + 2*x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 1, x^10 + 2*x^9 + 2*x^8 + x^7 + 2*x^6 + 2*x^2 + 2*x + 2, x^10 + x^8 + 2*x^7 + 2*x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2, 2*x^7 + x^5 + 2*x^3 + x^2 + x, 2*x^9 + 2*x^8 + x^7 + x^5 + x^4 + x^3 + x^2 + 2*x + 1, x^9 + 2*x^7 + 2*x^3 + x^2 + x + 2, x^10 + x^6 + x^5 + x^3 + x^2 + 2*x + 2, x^10 + 2*x^9 + 2*x^8 + 2*x^6 + 2*x^3 + 2*x^2 + 2*x + 2, 2*x^10 + x^7 + 2*x^6 + x^4 + 2*x^3 + x^2 [13]: [126, 12, 69] Linear Code over GF(3) Apply ConstructionXChain to [12] [11] [10] and [9] then apply ConstructionX using [8] last modified: 2010-01-04
Lb(126,12) = 68 is found by truncation of: Lb(127,12) = 69 GW2 Ub(126,12) = 75 LP
LP: Follows from the linear programming bound.
Notes
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