lower bound: | 143 |
upper bound: | 147 |
Construction of a linear code [228,9,143] 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]: [224, 7, 143] Quasicyclic of degree 4 Linear Code over GF(3) QuasiCyclicCode of length 224 with generating polynomials: x^54 + x^53 + 2*x^51 + 2*x^50 + 2*x^49 + 2*x^48 + x^47 + x^46 + x^45 + 2*x^43 + 2*x^41 + 2*x^40 + x^39 + x^38 + 2*x^36 + 2*x^34 + 2*x^32 + x^31 + 2*x^30 + x^29 + 2*x^28 + x^27 + x^26 + 2*x^24 + 2*x^23 + 2*x^21 + x^18 + 2*x^16 + 2*x^15 + 2*x^14 + 2*x^13 + x^10 + x^9 + x^7 + x^5, x^52 + x^51 + x^50 + 2*x^48 + x^47 + x^46 + 2*x^43 + x^42 + x^39 + 2*x^38 + x^35 + x^32 + x^30 + x^29 + x^28 + 2*x^27 + x^26 + 2*x^25 + x^23 + 2*x^21 + 2*x^20 + x^19 + 2*x^17 + x^16 + 2*x^13 + x^12 + x^11 + 2*x^10 + 2*x^9 + x^8 + x^7 + x^6 + 2*x^5 + 2*x^3 + x, x^54 + 2*x^53 + 2*x^52 + x^51 + 2*x^50 + x^49 + x^47 + 2*x^46 + x^45 + 2*x^44 + x^42 + 2*x^41 + x^40 + 2*x^39 + 2*x^38 + x^37 + 2*x^35 + 2*x^34 + 2*x^33 + x^32 + x^31 + 2*x^30 + 2*x^29 + 2*x^27 + 2*x^24 + x^23 + 2*x^22 + x^21 + x^20 + x^19 + 2*x^18 + x^17 + 2*x^16 + 2*x^15 + 2*x^10 + x^9 + x^8 + 2*x^6 + x^3 + 2*x^2 + 1, 2*x^55 + x^54 + 2*x^53 + 2*x^52 + x^50 + 2*x^49 + x^48 + x^47 + 2*x^46 + 2*x^45 + x^44 + 2*x^43 + 2*x^42 + 2*x^41 + 2*x^40 + x^39 + 2*x^38 + x^36 + x^35 + x^34 + x^31 + 2*x^29 + 2*x^27 + x^26 + 2*x^25 + x^23 + x^22 + 2*x^21 + x^20 + 2*x^17 + x^16 + 2*x^15 + 2*x^13 + x^9 + x^8 + x^6 + x^5 + 2*x^4 + 2*x^2 + 2*x + 2 [11]: [224, 8, 141] Quasicyclic of degree 4 Linear Code over GF(3) QuasiCyclicCode of length 224 stacked to height 2 with generating polynomials: x^55 + 2*x^54 + 2*x^53 + x^52 + x^51 + 2*x^50 + x^49 + 2*x^47 + 2*x^46 + x^45 + x^44 + x^43 + x^42 + 2*x^40 + 2*x^39 + 2*x^38 + 2*x^37 + 2*x^35 + 2*x^34 + 2*x^33 + 2*x^32 + x^31 + x^30 + x^28 + 2*x^26 + 2*x^25 + 2*x^22 + x^20 + 2*x^19 + 2*x^18 + x^13 + 2*x^12 + 2*x^11 + 2*x^10 + 2*x^9 + x^8 + 2*x^7 + 2*x^6 + 2*x^5 + 2*x^4 + x, 2*x^55 + x^54 + x^51 + x^50 + x^49 + x^48 + x^45 + x^43 + x^42 + 2*x^39 + x^38 + 2*x^36 + 2*x^35 + 2*x^33 + 2*x^31 + 2*x^30 + x^29 + x^28 + 2*x^27 + x^25 + x^24 + x^19 + x^18 + x^16 + x^13 + x^12 + 2*x^11 + x^9 + 2*x^8 + 2*x^7 + x^6 + 2*x^5 + x^4 + 2*x^3 + 2*x^2, x^55 + x^54 + 2*x^53 + x^52 + x^50 + x^49 + x^45 + x^44 + 2*x^43 + 2*x^41 + 2*x^40 + x^38 + 2*x^35 + 2*x^34 + 2*x^33 + x^32 + x^31 + 2*x^30 + 2*x^28 + x^27 + x^26 + x^24 + 2*x^23 + x^22 + x^21 + 2*x^20 + 2*x^19 + 2*x^18 + x^17 + x^16 + 2*x^14 + 2*x^11 + x^10 + 2*x^9 + 2*x^8 + x^4 + x^3 + 2*x, 2*x^55 + 2*x^54 + 2*x^52 + x^51 + x^50 + 2*x^49 + x^47 + 2*x^44 + 2*x^42 + 2*x^41 + x^40 + 2*x^38 + 2*x^37 + x^36 + x^34 + x^31 + 2*x^29 + 2*x^28 + x^27 + x^26 + x^24 + 2*x^23 + 2*x^22 + x^21 + 2*x^19 + x^16 + x^14 + x^13 + 2*x^12 + x^10 + x^9 + 2*x^8 + 2*x^6 + 2*x^3 + x + 1, x^54 + 2*x^53 + 2*x^52 + x^50 + x^48 + x^47 + x^45 + 2*x^44 + 2*x^43 + x^40 + 2*x^38 + 2*x^37 + x^35 + 2*x^34 + x^33 + x^32 + x^30 + 2*x^27 + x^26 + 2*x^23 + x^22 + 2*x^21 + x^20 + x^19 + 2*x^18 + x^17 + 2*x^14 + 2*x^13 + x^12 + 2*x^11 + 2*x^8 + x^7 + x^5 + x^4 + 2*x^3 + x^2 + 2*x + 2, x^55 + 2*x^53 + 2*x^51 + 2*x^49 + 2*x^46 + x^45 + x^44 + 2*x^43 + 2*x^42 + 2*x^40 + 2*x^39 + x^38 + 2*x^35 + 2*x^34 + 2*x^32 + 2*x^31 + x^29 + 2*x^28 + x^26 + 2*x^25 + x^24 + 2*x^23 + x^22 + 2*x^21 + x^20 + x^19 + 2*x^18 + 2*x^15 + 2*x^14 + x^13 + 2*x^12 + 2*x^11 + x^9 + x^8 + 2*x^7 + 2*x^6 + x^5 + 2*x^4 + 2*x^3 + x^2 + 2, x^55 + 2*x^54 + x^53 + 2*x^52 + 2*x^51 + 2*x^49 + 2*x^48 + 2*x^45 + 2*x^44 + x^43 + x^42 + 2*x^41 + 2*x^40 + x^39 + 2*x^36 + x^35 + x^33 + 2*x^32 + 2*x^31 + 2*x^30 + 2*x^29 + 2*x^27 + x^26 + 2*x^25 + x^24 + x^23 + x^21 + x^20 + x^17 + x^16 + 2*x^15 + 2*x^14 + x^13 + x^12 + 2*x^11 + x^8 + 2*x^7 + 2*x^5 + x^4 + x^3 + x^2 + x, x^54 + x^53 + x^51 + 2*x^50 + x^49 + 2*x^47 + x^46 + 2*x^45 + 2*x^43 + 2*x^41 + x^40 + 2*x^39 + 2*x^36 + x^32 + x^31 + 2*x^30 + 2*x^29 + 2*x^28 + x^27 + x^24 + 2*x^22 + x^20 + 2*x^19 + 2*x^17 + x^16 + 2*x^15 + x^14 + 2*x^13 + 2*x^11 + x^10 + x^9 + 2*x^8 + x^7 + x^6 + x^5 + 2*x^2 + 2*x + 2 [12]: [224, 9, 140] Quasicyclic of degree 4 Linear Code over GF(3) QuasiCyclicCode of length 224 stacked to height 2 with generating polynomials: 2*x^54 + 2*x^49 + 2*x^48 + 2*x^47 + x^45 + 2*x^44 + 2*x^43 + x^40 + 2*x^39 + 2*x^36 + x^35 + 2*x^32 + 2*x^29 + 2*x^27 + 2*x^26 + 2*x^25 + x^24 + 2*x^23 + x^22 + 2*x^20 + x^18 + x^17 + 2*x^16 + x^14 + 2*x^13 + x^10 + 2*x^9 + 2*x^8 + x^7 + x^6 + 2*x^5 + 2*x^4 + 2*x^3 + x^2 + 1, x^55 + x^54 + 2*x^53 + 2*x^52 + 2*x^51 + x^50 + 2*x^49 + 2*x^48 + x^47 + x^46 + x^44 + 2*x^43 + x^42 + 2*x^41 + x^40 + x^39 + x^37 + x^35 + x^34 + x^32 + x^30 + 2*x^29 + x^28 + x^27 + 2*x^26 + x^24 + 2*x^22 + x^20 + x^19 + 2*x^18 + 2*x^17 + 2*x^16 + 2*x^14 + 2*x^12 + x^11 + x^9 + x^7 + 2*x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 2, x^54 + x^52 + 2*x^51 + 2*x^50 + 2*x^47 + 2*x^46 + x^45 + x^39 + x^38 + 2*x^37 + x^34 + 2*x^33 + 2*x^32 + 2*x^30 + x^28 + 2*x^27 + 2*x^26 + 2*x^25 + 2*x^24 + x^22 + 2*x^21 + x^18 + x^17 + 2*x^15 + 2*x^13 + x^11 + 2*x^10 + 2*x^7 + 2*x^6 + x^4 + 2*x^3 + x^2 + 2*x + 2, x^55 + 2*x^52 + x^49 + x^47 + 2*x^46 + x^45 + 2*x^44 + 2*x^43 + x^42 + x^41 + x^40 + x^39 + x^37 + x^34 + x^31 + 2*x^29 + x^25 + x^24 + x^23 + x^18 + x^16 + 2*x^15 + 2*x^14 + 2*x^12 + x^7 + 2*x^6 + x^5 + 2*x, x^54 + x^53 + x^52 + x^50 + 2*x^49 + x^48 + x^46 + 2*x^45 + 2*x^43 + 2*x^42 + x^41 + x^40 + 2*x^39 + 2*x^38 + 2*x^37 + 2*x^36 + 2*x^35 + x^33 + 2*x^32 + 2*x^31 + x^30 + 2*x^29 + 2*x^25 + x^21 + x^17 + x^16 + x^15 + 2*x^14 + 2*x^13 + x^11 + 2*x^10 + x^9 + 2*x^8 + x^7 + x^6 + 2*x^5 + 2*x^4 + x^3 + x + 1, 2*x^54 + 2*x^53 + 2*x^51 + 2*x^49 + 2*x^42 + 2*x^41 + x^39 + x^38 + x^37 + x^36 + 2*x^35 + 2*x^34 + 2*x^33 + x^31 + x^29 + x^28 + 2*x^27 + 2*x^26 + x^24 + x^22 + x^20 + 2*x^19 + x^18 + 2*x^17 + x^16 + 2*x^15 + 2*x^14 + x^12 + x^11 + x^9 + 2*x^6 + x^4 + x^3 + x^2 + x, 2*x^53 + 2*x^50 + x^49 + x^48 + x^47 + x^45 + x^44 + 2*x^43 + 2*x^42 + 2*x^41 + x^40 + 2*x^39 + x^38 + 2*x^37 + 2*x^36 + 2*x^35 + x^34 + x^30 + 2*x^28 + x^27 + 2*x^25 + x^23 + x^22 + 2*x^20 + 2*x^16 + 2*x^15 + x^14 + 2*x^13 + 2*x^12 + 2*x^11 + 2*x^10 + 2*x^9 + x^8 + 2*x^7 + 2*x^6 + x^5 + x^3 + 2*x^2 + x + 1, x^55 + 2*x^54 + x^53 + 2*x^52 + x^50 + 2*x^49 + 2*x^48 + x^47 + x^46 + 2*x^45 + x^43 + x^41 + x^40 + 2*x^39 + 2*x^37 + x^35 + x^34 + 2*x^33 + 2*x^32 + x^31 + x^30 + 2*x^28 + 2*x^27 + 2*x^26 + 2*x^25 + 2*x^24 + x^21 + 2*x^20 + 2*x^19 + x^17 + x^16 + 2*x^15 + x^14 + 2*x^13 + x^11 + x^10 + x^9 + x^8 + 2*x^7 + x^5 + 2*x^4 + 2*x^3 + 2 [13]: [229, 9, 144] Linear Code over GF(3) ConstructionXX using [12] [11] [10] [9] and [8] [14]: [228, 9, 143] Linear Code over GF(3) Puncturing of [13] at { 229 } last modified: 2008-11-06
Lb(228,9) = 140 is found by taking a subcode of: Lb(228,10) = 140 MSY Ub(228,9) = 147 follows by a one-step Griesmer bound from: Ub(80,8) = 49 follows by a one-step Griesmer bound from: Ub(30,7) = 16 is found by considering shortening to: Ub(29,6) = 16 is found by considering truncation to: Ub(28,6) = 15 HHM
MSY: T. Maruta, M. Shinohara, F. Yamane, K. Tsuji, E. Takata, H. Miki & R. Fujiwara, New linear codes from cyclic or generalized cyclic codes by puncturing, to appear in Proc. 10th International Workshop on Algebraic and Combinatorial Coding Theory(ACCT-10) in Zvenigorod, Russia, 2006.
Notes
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