lower bound: | 78 |
upper bound: | 96 |
Construction of a linear code [177,26,78] over GF(3): [1]: [12, 6, 6] Linear Code over GF(3) Extend the QRCode over GF(3)of length 11 [2]: [10, 4, 6] Linear Code over GF(3) Shortening of [1] at { 11 .. 12 } [3]: [121, 116, 3] "Hamming code (r = 5)" Linear Code over GF(3) 5-th order HammingCode over GF( 3) [4]: [118, 113, 3] Linear Code over GF(3) Shortening of [3] at { 119 .. 121 } [5]: [40, 36, 3] Linear Code over GF(3) Construction B of [4] [6]: [37, 33, 3] Linear Code over GF(3) Shortening of [5] at { 38 .. 40 } [7]: [13, 10, 3] Linear Code over GF(3) Construction B of [6] [8]: [7, 4, 3] Linear Code over GF(3) Shortening of [7] at { 8 .. 13 } [9]: [160, 22, 75] Quasicyclic of degree 2 Linear Code over GF(3) QuasiCyclicCode of length 160 with generating polynomials: x^77 + x^76 + 2*x^74 + 2*x^73 + x^72 + x^71 + x^70 + 2*x^68 + 2*x^66 + x^65 + 2*x^64 + x^59 + x^56 + x^54 + 2*x^53 + x^52 + 2*x^50 + x^49 + 2*x^48 + x^46 + 2*x^45 + 2*x^44 + x^40 + x^39 + x^38 + 2*x^37 + x^36 + 2*x^34 + x^33 + 2*x^31 + 2*x^30 + x^28 + 2*x^27 + x^23 + x^22 + x^7, x^77 + x^76 + 2*x^75 + x^74 + 2*x^71 + x^70 + 2*x^68 + x^67 + 2*x^65 + x^64 + 2*x^62 + 2*x^61 + 2*x^60 + 2*x^59 + 2*x^58 + 2*x^57 + 2*x^56 + x^53 + x^52 + 2*x^51 + x^50 + x^49 + x^48 + x^47 + 2*x^46 + 2*x^45 + 2*x^43 + 2*x^41 + 2*x^40 + 2*x^39 + 2*x^38 + 2*x^37 + 2*x^36 + x^33 + x^32 + x^31 + x^30 + x^29 + x^27 + 2*x^26 + x^25 + 2*x^24 + x^22 + x^15 + 2*x^14 + x^12 + x^11 + x^10 + 2*x^8 + x^7 + x^5 + 2*x^3 + 2*x + 1 [10]: [160, 22, 72] Quasicyclic of degree 2 Linear Code over GF(3) QuasiCyclicCode of length 160 with generating polynomials: x^79 + x^78 + 2*x^77 + 2*x^76 + 2*x^74 + 2*x^73 + x^72 + 2*x^71 + x^70 + 2*x^67 + x^63 + x^62 + x^61 + 2*x^55 + x^53 + 2*x^52 + x^51 + x^50 + 2*x^49 + x^48 + 2*x^46 + 2*x^44 + 2*x^43 + 2*x^42 + x^41 + x^40 + 2*x^38 + x^36 + x^34 + x^33 + 2*x^32 + 2*x^28 + 2*x^27 + x^26 + x^25 + x^24 + 2*x^23 + x^22 + x, 2*x^77 + 2*x^76 + x^75 + 2*x^74 + x^72 + x^68 + 2*x^67 + 2*x^66 + 2*x^65 + x^64 + 2*x^61 + 2*x^58 + 2*x^55 + x^54 + x^51 + x^50 + x^49 + 2*x^48 + 2*x^47 + x^46 + 2*x^44 + x^43 + 2*x^42 + 2*x^39 + 2*x^38 + x^37 + x^36 + x^35 + 2*x^33 + 2*x^31 + x^28 + 2*x^26 + x^25 + 2*x^23 + 2*x^19 + 2*x^18 + 2*x^17 + 2*x^16 + 2*x^14 + 2*x^13 + x^11 + x^9 + 2*x^8 + 2*x^6 + 2*x^5 + x^4 + 2*x^3 + 2 [11]: [160, 26, 69] Quasicyclic of degree 2 Linear Code over GF(3) QuasiCyclicCode of length 160 with generating polynomials: 2*x^79 + x^78 + 2*x^76 + x^75 + x^73 + 2*x^72 + x^71 + 2*x^70 + x^69 + x^67 + 2*x^66 + x^65 + 2*x^63 + x^62 + x^61 + 2*x^60 + x^58 + x^57 + 2*x^56 + 2*x^55 + 2*x^53 + 2*x^52 + 2*x^51 + 2*x^50 + 2*x^49 + x^48 + x^47 + x^46 + x^45 + 2*x^43 + x^40 + 2*x^39 + 2*x^36 + 2*x^33 + x^32 + 2*x^31 + 2*x^30 + 2*x^29 + x^25, x^77 + x^76 + 2*x^73 + 2*x^72 + x^71 + 2*x^70 + 2*x^69 + x^68 + x^67 + x^63 + 2*x^62 + 2*x^61 + x^60 + 2*x^58 + 2*x^57 + 2*x^56 + 2*x^54 + x^53 + 2*x^51 + 2*x^49 + 2*x^47 + x^45 + 2*x^44 + 2*x^39 + x^38 + 2*x^37 + 2*x^36 + x^35 + x^33 + 2*x^30 + x^29 + 2*x^28 + x^27 + 2*x^26 + 2*x^25 + x^24 + 2*x^22 + x^18 + 2*x^17 + x^15 + x^14 + 2*x^13 + 2*x^10 + 2*x^9 + 2*x^8 + x^7 + 2*x^6 + x^5 + 2*x^3 + x + 2 [12]: [177, 26, 78] Linear Code over GF(3) ConstructionXX using [11] [10] [9] [8] and [2] last modified: 2021-08-26
Lb(177,26) = 75 is found by truncation of: Lb(180,26) = 78 BZ Ub(177,26) = 96 is found by considering shortening to: Ub(175,24) = 96 Da2
Da2: R.N. Daskalov, The linear programming bound for ternary linear codes, p. 423 in: IEEE International Symposium on Information Theory, Trondheim, 1994.
Notes
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