lower bound: | 106 |
upper bound: | 109 |
Construction of a linear code [128,5,106] over GF(8): [1]: [1, 1, 1] Cyclic Linear Code over GF(2^3) RepetitionCode of length 1 [2]: [126, 4, 105] Quasicyclic of degree 9 Linear Code over GF(2^3) QuasiCyclicCode of length 126 with generating polynomials: w^2*x^13 + w^4*x^12 + w^5*x^11 + w^6*x^10 + w^2*x^9 + w^5*x^8 + w^3*x^7 + w*x^6 + w^5*x^5 + w^2*x^4 + 1, w^5*x^13 + w*x^12 + w^6*x^11 + w^2*x^10 + x^9 + w^6*x^7 + x^6 + w^3*x^5 + w^4*x^4 + x^3 + w^6*x^2 + w^5*x + w^5, w^4*x^13 + w^5*x^12 + w*x^11 + w*x^10 + w*x^9 + w^2*x^8 + w^5*x^7 + w^4*x^5 + x^4 + w^5*x^3 + w^4*x^2 + w^3*x + w^6, x^13 + w^6*x^12 + w^3*x^11 + w^6*x^10 + w^2*x^9 + w^4*x^8 + w^4*x^7 + w^4*x^5 + w^4*x^4 + x^3 + w^3*x^2 + w^2*x, w*x^13 + x^12 + x^11 + w^2*x^10 + x^9 + x^8 + w^3*x^6 + w*x^5 + w^2*x^4 + w^4*x^2 + w^3*x + w^4, w^5*x^13 + w^4*x^12 + w^5*x^11 + w^5*x^10 + w^6*x^9 + w^3*x^8 + x^7 + w^4*x^6 + w^6*x^5 + w*x^4 + w^3*x^3 + w*x^2 + x + w^5, w^5*x^13 + w^3*x^12 + w^4*x^10 + w*x^8 + w^2*x^7 + w*x^6 + w^5*x^5 + w^5*x^4 + w^2*x^3 + w^4*x^2 + w^3*x + w^5, w^2*x^13 + w^5*x^12 + w*x^11 + w^5*x^10 + w^6*x^9 + w^6*x^8 + w*x^7 + x^6 + w^3*x^5 + w^6*x^4 + w^6*x^3 + w^3*x^2 + w^2*x + 1, x^13 + w^2*x^11 + w^2*x^10 + x^9 + w^5*x^8 + w^3*x^7 + w^2*x^6 + w^2*x^5 + w^3*x^4 + w^4*x^3 + w^5*x^2 + w^4*x [3]: [126, 4, 105] Quasicyclic of degree 9 Linear Code over GF(2^3) QuasiCyclicCode of length 126 with generating polynomials: w^2*x^13 + w^4*x^12 + w^3*x^11 + x^10 + w^6*x^9 + w^4*x^8 + w^4*x^7 + w^5*x^6 + w*x^5 + x^4 + x, w^4*x^13 + w^4*x^11 + w^6*x^10 + w^3*x^9 + w^5*x^8 + w*x^7 + w^3*x^5 + w*x^4 + w^5*x^3 + w*x^2 + w*x + w^6, w^4*x^13 + w^2*x^12 + w^4*x^11 + w^2*x^10 + w^3*x^9 + w*x^7 + w^3*x^6 + w^3*x^5 + w^5*x^3 + w^5*x^2 + w*x + w^3, w^2*x^13 + w*x^12 + w^2*x^11 + w*x^10 + w^6*x^8 + w^3*x^7 + w^6*x^4 + w^5*x^3 + w^5*x^2 + w^3*x, w^2*x^13 + w^6*x^12 + w^2*x^11 + x^10 + w^5*x^8 + w^3*x^7 + w^6*x^6 + w^3*x^4 + w^5*x^3 + w^3*x^2 + w^3*x + 1, w*x^13 + w^2*x^12 + w^4*x^10 + x^9 + w^2*x^8 + w^2*x^7 + w^5*x^6 + w^3*x^5 + w^4*x^4 + w^6*x^3 + w^6*x^2 + w^4*x + w^6, w*x^13 + w^4*x^12 + w^2*x^11 + w^2*x^10 + w*x^9 + w^5*x^8 + w^5*x^7 + w^4*x^6 + w^2*x^5 + w^6*x^4 + x^3 + w^6*x^2 + x + w^2, w^5*x^13 + w^4*x^12 + w*x^11 + w^2*x^10 + w^4*x^9 + w^5*x^8 + w^4*x^7 + w^4*x^6 + w^3*x^5 + w^6*x^4 + w*x^3 + w^6*x^2 + w^3*x + w^2, w^6*x^13 + w^6*x^12 + w^5*x^11 + w^4*x^10 + w*x^8 + w*x^7 + w*x^5 + x^4 + w^6*x^3 + w^2*x^2 + w^3 [4]: [126, 5, 104] Quasicyclic of degree 9 Linear Code over GF(2^3) QuasiCyclicCode of length 126 with generating polynomials: x^13 + w^3*x^12 + w^3*x^11 + x^10 + x^9 + w*x^8 + w*x^7 + w^3*x^6 + w^3*x^5 + 1, w^2*x^13 + w^6*x^11 + w^3*x^10 + w^6*x^9 + w^5*x^8 + w^3*x^7 + w^6*x^6 + w^2*x^5 + w^2*x^4 + w^3*x^3 + x^2 + w*x + w^4, w^3*x^13 + w^3*x^12 + w^4*x^11 + x^10 + w^5*x^9 + w*x^8 + w^4*x^7 + w^3*x^6 + w*x^5 + w^5*x^3 + w^3*x + 1, w*x^13 + w^2*x^12 + x^11 + w^2*x^9 + w^2*x^8 + w^2*x^7 + w*x^6 + w^5*x^5 + x^3 + w^4*x^2 + w^5*x + w^4, x^12 + w^2*x^11 + w^4*x^10 + w*x^9 + w^3*x^6 + w^4*x^5 + w^5*x^4 + w^4*x^3 + w^6*x^2 + w^2*x + w^2, w^2*x^13 + w^2*x^12 + w^6*x^11 + w^6*x^9 + w^2*x^8 + w^3*x^7 + w*x^6 + w^2*x^5 + w^3*x^3 + w^4*x^2 + w*x + w^4, w^5*x^13 + w^6*x^12 + w^5*x^11 + w*x^8 + w^6*x^7 + w^6*x^6 + w^5*x^4 + w*x^3 + w^5*x^2 + w^6*x, w^4*x^13 + w^2*x^12 + w^5*x^10 + w^3*x^9 + x^8 + w^5*x^7 + w^5*x^6 + w^6*x^5 + w*x^4 + w^2*x^3 + x^2 + x + w^2, w^5*x^13 + w^4*x^12 + w^5*x^11 + w*x^10 + w^5*x^8 + w^6*x^7 + w^3*x^6 + w^3*x^4 + w*x^3 + w^4*x^2 + w^6*x + w^5 [5]: [128, 5, 106] Linear Code over GF(2^3) ConstructionXX using [4] [3] [2] [1] and [1] last modified: 2008-11-06
Lb(128,5) = 105 is found by truncation of: Lb(130,5) = 107 BKW Ub(128,5) = 109 follows by a one-step Griesmer bound from: Ub(18,4) = 13 is found by considering shortening to: Ub(17,3) = 13 is found by considering truncation to: Ub(16,3) = 12 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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