lower bound: | 156 |
upper bound: | 159 |
Construction of a linear code [243,7,156] over GF(3): [1]: [12, 6, 6] Linear Code over GF(3) Extend the QRCode over GF(3)of length 11 [2]: [11, 6, 5] Cyclic Linear Code over GF(3) Puncturing of [1] at { 12 } [3]: [9, 4, 5] Linear Code over GF(3) Shortening of [2] at { 10 .. 11 } [4]: [234, 3, 162] Quasicyclic of degree 18 Linear Code over GF(3) QuasiCyclicCode of length 234 with generating polynomials: 2*x^12 + x^11 + 2*x^9 + 2*x^8 + x^7 + x^6 + x^5 + x^3 + 1, x^12 + 2*x^10 + 2*x^9 + x^8 + x^7 + x^6 + x^4 + x + 2, 2*x^12 + 2*x^11 + x^10 + x^9 + x^8 + x^6 + x^3 + 2*x^2 + x, x^12 + 2*x^11 + x^10 + 2*x^8 + 2*x^7 + x^6 + x^5 + x^4 + x^2, 2*x^12 + 2*x^11 + 2*x^9 + 2*x^6 + x^5 + 2*x^4 + x^2 + x + 2, x^12 + 2*x^11 + x^9 + x^8 + 2*x^7 + 2*x^6 + 2*x^5 + 2*x^3 + 2, x^12 + x^10 + x^7 + 2*x^6 + x^5 + 2*x^3 + 2*x^2 + x + 1, 2*x^12 + 2*x^11 + 2*x^10 + 2*x^8 + 2*x^5 + x^4 + 2*x^3 + x + 1, 2*x^11 + x^10 + 2*x^9 + x^7 + x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x, x^11 + x^8 + 2*x^7 + x^6 + 2*x^4 + 2*x^3 + x^2 + x + 1, 2*x^12 + 2*x^10 + 2*x^7 + x^6 + 2*x^5 + x^3 + x^2 + 2*x + 2, 2*x^10 + x^9 + 2*x^8 + x^6 + x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 2, x^12 + 2*x^11 + 2*x^10 + 2*x^9 + 2*x^7 + 2*x^4 + x^3 + 2*x^2 + 1, x^12 + x^10 + x^7 + 2*x^6 + x^5 + 2*x^3 + 2*x^2 + x + 1, x^12 + x^11 + x^10 + x^8 + x^5 + 2*x^4 + x^3 + 2*x + 2, 2*x^11 + 2*x^10 + x^9 + x^8 + x^7 + x^5 + x^2 + 2*x + 1, 2*x^11 + 2*x^8 + x^7 + 2*x^6 + x^4 + x^3 + 2*x^2 + 2*x + 2, 2*x^12 + x^11 + 2*x^10 + x^8 + x^7 + 2*x^6 + 2*x^5 + 2*x^4 + 2*x^2 [5]: [234, 7, 151] Quasicyclic of degree 18 Linear Code over GF(3) QuasiCyclicCode of length 234 with generating polynomials: x^12 + x^9 + x^7 + 1, x^12 + x^11 + x^6 + 2*x^5 + 2*x^4 + x^3 + x^2 + x + 2, 2*x^12 + x^10 + x^9 + x^8 + 2*x^7 + x^6 + x^5 + x^4 + x^3 + 2, x^12 + x^10 + x^9 + 2*x^7 + x^6 + 2*x^5 + 2*x^4 + 2*x^3 + x^2 + 1, 2*x^12 + x^10 + 2*x^9 + 2*x^8 + x^7 + 2*x^5 + x^4 + x^3 + 2*x + 2, 2*x^10 + 2*x^9 + 2*x^8 + x^4 + x^3 + 2*x^2 + x, x^11 + x^10 + x^9 + 2*x^8 + x^7 + x^4 + 2*x^2 + 2*x + 1, x^12 + 2*x^11 + 2*x^10 + x^8 + x^6 + x^5 + 2*x^4 + 2*x^2 + x + 1, x^11 + x^10 + 2*x^5 + 2*x^3 + x^2 + 2*x + 1, 2*x^10 + x^9 + 2*x^7 + 2*x^5 + x^2 + x, x^12 + 2*x^11 + x^10 + 2*x^9 + x^3 + 2*x, 2*x^12 + 2*x^11 + x^10 + x^9 + 2*x^8 + 2*x^7 + x^6 + 2*x^5 + x^4 + 2*x^3 + 2*x^2 + 2*x + 2, 2*x^12 + x^11 + 2*x^9 + 2*x^8 + 2*x^7 + x^6 + 2*x^3 + 2*x^2 + x, 2*x^11 + x^10 + 2*x^7 + 2*x^6 + 2*x^5 + x^4 + 2*x^3 + x^2 + x + 2, x^11 + x^9 + 2*x^7 + x^6 + 2*x^5 + 2*x^3 + 2, 2*x^11 + x^10 + x^8 + x^7 + x^6 + 2*x^4 + 2*x, 2*x^11 + x^10 + 2*x^8 + x^6 + 2*x^5 + x^3 + 2, x^12 + x^11 + x^9 + 2*x^8 + x^7 + x^6 + 2*x^3 + 2*x^2 + x + 2 [6]: [243, 7, 156] Linear Code over GF(3) ConstructionX using [5] [4] and [3] last modified: 2003-10-08
Lb(243,7) = 156 Koh Ub(243,7) = 159 follows by a one-step Griesmer bound from: Ub(83,6) = 53 is found by considering truncation to: Ub(81,6) = 51 Ma
Ma: T. Maruta, On the nonexistence of linear codes attaining the Griesmer bound, Geom. Dedicata 60 (1996) 1-7. T. Maruta, On the nonexistence of linear codes of dimension four attaining the Griesmer bound, pp. 117-120 in: Optimal codes and related topics, Proc. Workshop Sozopol, Bulgaria, 1995. T. Maruta, The nonexistence of [116,5,85]_4 codes and [187,5,139]_4 codes, Proc. 2nd International Workshop on Optimal Codes and Related Topics in Sozopol (1998), pp. 168-174. T. Maruta & M. Fukui, On the nonexistence of some linear codes of dimension 4 over GF(5), preprint, 1995. T. Maruta, M. Takenaka, M. Shinohara, K. Masuda & S. Kawashima, Constructing new linear codes over small fields, preprint 2004.
Notes
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