lower bound: | 75 |
upper bound: | 81 |
Construction of a linear code [132,11,75] 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]: [128, 9, 75] Quasicyclic of degree 8 Linear Code over GF(3) QuasiCyclicCode of length 128 with generating polynomials: x^13 + x^12 + x^11 + 2*x^9 + x^4, 2*x^15 + x^14 + 2*x^13 + 2*x^12 + 2*x^11 + x^10 + 2*x^9 + 2*x^8 + x^7 + 2*x^5 + x^4 + x^3 + 2*x^2 + x + 2, x^15 + 2*x^14 + x^13 + x^12 + 2*x^10 + x^9 + x^8 + 2*x^4 + x^3 + x^2 + x + 1, x^15 + x^14 + x^9 + x^6 + x^5 + 2*x^4 + 2*x^2 + 2*x + 1, x^15 + 2*x^14 + x^12 + x^11 + x^10 + 2*x^7 + x^4 + 2*x^3 + 2*x^2 + x + 1, 2*x^15 + x^14 + x^13 + 2*x^12 + x^11 + 2*x^9 + 2*x^8 + 2*x^7 + 2*x^4 + 2*x^2 + 1, 2*x^14 + x^13 + 2*x^11 + 2*x^10 + x^9 + 2*x^8 + x^7 + 2*x^5 + x^3 + x, x^15 + 2*x^14 + 2*x^12 + 2*x^10 + 2*x^9 + 2*x^8 + 2*x^6 + x^4 + 2*x^3 + x^2 + x [10]: [128, 11, 72] Quasicyclic of degree 8 Linear Code over GF(3) QuasiCyclicCode of length 128 stacked to height 2 with generating polynomials: 0, x^15 + 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^15 + 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, 2*x^15 + 2*x^14 + 2*x^13 + 2*x^12 + 2*x^11 + 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^15 + 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, 2*x^15 + 2*x^14 + 2*x^13 + 2*x^12 + 2*x^11 + 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^15 + 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^15 + x^14 + x^12 + x^11 + 2*x^10 + x^9, x^15 + 2*x^14 + x^12 + x^10 + 2*x^8 + x^7 + 2*x^6 + x^5 + 2*x^4 + x^3 + 2*x^2 + 2*x + 1, x^15 + 2*x^14 + 2*x^13 + 2*x^12 + x^9 + 2*x^7 + x^6 + 2*x^5 + x^4 + x^3 + x^2 + 2*x, x^14 + 2*x^13 + 2*x^12 + x^11 + 2*x^10 + x^8 + 2*x^7 + x^6 + 2*x^4 + x^3 + 2*x^2 + x + 1, 2*x^15 + x^14 + x^12 + 2*x^11 + 2*x^10 + 2*x^9 + 2*x^8 + 2*x^7 + x^5 + x + 2, x^15 + 2*x^14 + 2*x^10 + 2*x^9 + x^8 + 2*x^7 + 2*x^6 + x^4 + x^3, x^15 + 2*x^14 + 2*x^12 + 2*x^11 + x^8 + 2*x^7 + 2*x^6 + x^5 + 2*x^4 + x^3 + 1, 2*x^15 + x^14 + 2*x^12 + x^11 + x^8 + x^5 + x^4 + x^2 + 1 [11]: [132, 11, 75] Linear Code over GF(3) ConstructionX using [10] [9] and [8] last modified: 2010-01-04
Lb(132,11) = 73 is found by truncation of: Lb(134,11) = 75 XX Ub(132,11) = 81 follows by a one-step Griesmer bound from: Ub(50,10) = 27 follows by a one-step Griesmer bound from: Ub(22,9) = 9 follows by a one-step Griesmer bound from: Ub(12,8) = 3 is found by considering shortening to: Ub(11,7) = 3 is found by construction B: [consider deleting the (at most) 6 coordinates of a word in the dual]
Notes
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