lower bound: | 67 |
upper bound: | 74 |
Construction of a linear code [168,18,67] over GF(2): [1]: [3, 2, 2] Cyclic Linear Code over GF(2) CordaroWagnerCode of length 3 [2]: [4, 1, 4] Cyclic Linear Code over GF(2) RepetitionCode of length 4 [3]: [4, 3, 2] Cyclic Linear Code over GF(2) Dual of the RepetitionCode of length 4 [4]: [8, 4, 4] Quasicyclic of degree 2 Linear Code over GF(2) PlotkinSum of [3] and [2] [5]: [8, 7, 2] Cyclic Linear Code over GF(2) Dual of the RepetitionCode of length 8 [6]: [16, 11, 4] Linear Code over GF(2) PlotkinSum of [5] and [4] [7]: [13, 8, 4] Linear Code over GF(2) Shortening of [6] at { 14 .. 16 } [8]: [153, 16, 64] Quasicyclic of degree 3 Linear Code over GF(2) QuasiCyclicCode of length 153 with generating polynomials: x^49 + x^48 + x^47 + x^44 + x^42 + x^41 + x^36 + x^35 + x^33 + x^32 + x^31 + x^30 + x^28 + x^27 + x^25 + x^24 + x^19 + x^18 + x^16 + x^11, x^50 + x^49 + x^48 + x^47 + x^46 + x^45 + x^44 + x^41 + x^39 + x^33 + x^30 + x^26 + x^25 + x^23 + x^19 + x^17 + x^15 + x^14 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^2 + 1, x^50 + x^49 + x^48 + x^47 + x^46 + x^45 + x^42 + x^40 + x^38 + x^37 + x^36 + x^34 + x^33 + x^30 + x^27 + x^26 + x^25 + x^22 + x^21 + x^19 + x^18 + x^17 + x^15 + x^14 + x^12 + x^11 + x^10 + x^9 + x^6 + x^5 + x^3 + x [9]: [153, 10, 70] Quasicyclic of degree 3 Linear Code over GF(2) QuasiCyclicCode of length 153 with generating polynomials: x^50 + x^48 + x^44 + x^43 + x^39 + x^37 + x^36 + x^35 + x^34 + x^31 + x^26 + x^22 + x^20 + x^19 + x^18 + x^16 + x^10 + 1, x^46 + x^45 + x^44 + x^43 + x^41 + x^37 + x^34 + x^33 + x^30 + x^28 + x^23 + x^22 + x^19 + x^18 + x^16 + x^13 + x^12 + x^10 + x^9 + x^7 + x^6 + x^5 + x^3 + x^2 + x + 1, x^50 + x^49 + x^48 + x^47 + x^44 + x^43 + x^38 + x^36 + x^35 + x^32 + x^30 + x^29 + x^27 + x^26 + x^24 + x^22 + x^21 + x^18 + x^17 + x^16 + x^14 + x^12 + x^7 + x^5 + x^2 + 1 [10]: [153, 18, 62] Quasicyclic of degree 3 Linear Code over GF(2) QuasiCyclicCode of length 153 with generating polynomials: x^50 + x^48 + x^47 + x^46 + x^45 + x^44 + x^41 + x^37 + x^36 + x^35 + x^33 + x^31 + x^30 + x^29 + x^28 + x^27 + x^26 + x^24 + x^20 + x^19 + x^18 + x^9, x^49 + x^48 + x^46 + x^45 + x^43 + x^42 + x^41 + x^38 + x^37 + x^36 + x^35 + x^34 + x^30 + x^25 + x^23 + x^22 + x^21 + x^18 + x^15 + x^14 + x^13 + x^12 + x^11 + x^9 + x^7 + x^6 + x^5 + x^3 + x^2 + 1, x^49 + x^42 + x^41 + x^40 + x^37 + x^35 + x^33 + x^32 + x^30 + x^29 + x^28 + x^26 + x^22 + x^19 + x^18 + x^16 + x^13 + x^12 + x^11 + x^9 + x^8 + x^7 + x^6 + x^5 + x^3 + x^2 [11]: [169, 18, 68] Linear Code over GF(2) ConstructionXX using [10] [9] [8] [7] and [1] [12]: [168, 18, 67] Linear Code over GF(2) Puncturing of [11] at { 169 } last modified: 2007-08-03
Lb(168,18) = 66 is found by taking a subcode of: Lb(168,19) = 66 GG1 Ub(168,18) = 74 is found by considering lengthening to: Ub(170,18) = 74 otherwise adding a parity check bit would contradict: Ub(171,18) = 75 BK
GG1: B. Groneick & S. Grosse, priv. comm. and comm. via W. Scharlau, 1992-1993.
Notes
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