lower bound: | 45 |
upper bound: | 48 |
Construction of a linear code [65,7,45] over GF(5): [1]: [1, 1, 1] Cyclic Linear Code over GF(5) RepetitionCode of length 1 [2]: [64, 6, 45] Linear Code over GF(5) QuasiCyclicCode of length 64 stacked to height 3 with generating polynomials: 3*x^3 + x^2 + x, x^3 + x^2 + 3, 4*x^3 + 3*x^2 + 3*x, x^2 + 2*x + 2, 2*x^3 + 2*x^2 + 1, 4*x^2 + 3*x + 3, 4*x^3 + 4*x^2 + 2, 4*x^3 + 4*x^2 + 2, 0, 2*x^3 + 3*x^2 + 2*x + 3, x^3 + 2*x^2 + 2*x, x^3 + 3*x^2 + 4*x + 2, 3*x^3 + 3*x^2 + 4, 4*x^3 + 2*x^2 + x + 3, 0, 3*x^3 + 2*x^2 + 3*x + 2, 4*x^3 + x^2, 4*x^3 + x^2, 2*x^2 + 3*x, 2*x^3 + 3, 3*x^3 + 4*x^2 + 3, 3*x^3 + x^2 + 1, 3*x^3 + x + 1, 3*x^3 + 4*x^2 + 3*x, 3*x^3 + 4*x^2 + 3*x, 2*x^2 + 3, 4*x^3 + x, 3*x^3 + x^2 + 1, 4*x^3 + 3*x^2 + x + 2, 4*x^3 + x^2 + 3*x + 2, 4*x^3 + 4*x^2 + 2*x, 4*x^3 + 4*x^2 + x + 1, 4*x^3 + 1, 3*x^3 + 2*x, 4*x^3 + x^2 + x + 4, x^3 + 3*x^2 + x, 3*x^3 + 2*x^2 + 2*x + 3, x^3 + x^2 + 2*x + 1, x^3 + 2*x^2 + 3*x + 4, x^2 + 4, 2*x^3 + x^2 + x + 1, 2*x^3 + 3*x^2, 2*x^3 + 2*x^2 + 2*x + 4, x^2 + 4*x, 2*x^3 + x^2 + 2*x, 2*x^3 + 3*x^2 + 4*x + 1, x^2 + 3*x + 1, x^3 + 3*x^2 + 2*x + 4 [3]: [64, 7, 44] Linear Code over GF(5) QuasiCyclicCode of length 64 stacked to height 3 with generating polynomials: x, x, x^3 + 4*x^2 + 3*x + 3, 3*x^3 + 4*x^2 + 4*x, 3*x^3 + x^2 + x + 1, 2*x^3 + 2*x^2 + 2*x, 4*x^2 + x + 1, 4*x^3 + 4*x^2 + x + 2, 3*x^3 + x^2 + 2*x, x^2 + 3*x + 2, 4*x^3 + 4*x + 3, 3*x^3 + x + 2, 3*x^3 + 3*x^2, 2*x + 4, x^2, 2*x^3 + 2*x^2 + x + 1, 2*x^3 + 1, 2*x^3 + 2*x^2 + 4*x, x^3 + 2*x^2 + 2*x + 3, 4*x^2 + 3*x + 1, x^3 + 3*x^2 + 2*x + 2, x^3 + 3*x^2 + 4*x, x^3 + x^2 + x, 3*x^3 + 3*x^2 + 4*x + 3, x^3 + 3*x^2 + 3*x + 1, 4*x^2 + 2*x + 2, 3, 3*x^3 + x^2 + x + 3, 4*x^3 + 3*x^2 + x, x^3 + 3*x + 4, 4*x^3 + x^2 + 3, 3*x^3 + 3*x + 2, x^3 + 1, 2*x^3 + 4*x^2 + x, 3*x^2 + 4, x^2 + x, 3*x^3 + 4*x^2 + 2*x + 3, x^3 + 4*x^2 + 2, 2*x^3 + x^2 + 4*x, 2*x^3 + 2*x^2 + 2*x + 1, x^3 + 2*x^2 + 4*x, 3*x^3 + 3*x^2 + 2*x + 4, 4*x + 3, x^3 + 2*x + 4, 2*x^3 + 3*x^2 + 2*x, 3*x^3 + 4*x^2 + x + 4, x^3 + 4*x + 2, 4*x^3 + 2*x^2 + 3*x + 3 [4]: [65, 7, 45] Linear Code over GF(5) ConstructionX using [3] [2] and [1] last modified: 2006-09-27
Lb(65,7) = 44 is found by truncation of: Lb(66,7) = 45 GW2 Ub(65,7) = 48 follows by a one-step Griesmer bound from: Ub(16,6) = 9 is found by considering shortening to: Ub(13,3) = 9 is found by considering truncation to: Ub(12,3) = 8 Hi4
Hi4: R. Hill, Optimal linear codes, pp. 75-104 in: Cryptography and Coding II (C. Mitchell, ed.), Oxford Univ. Press, 1992.
Notes
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