lower bound: | 20 |
upper bound: | 33 |
Construction of a linear code [109,63,20] over GF(4): [1]: [4, 3, 2] Cyclic Linear Code over GF(2^2) Dual of the RepetitionCode of length 4 [2]: [105, 60, 20] Constacyclic by w Linear Code over GF(2^2) ConstaCyclicCode generated by w*x^104 + w^2*x^103 + w*x^102 + x^101 + x^100 + w^2*x^99 + w*x^98 + w*x^97 + w^2*x^96 + x^95 + x^94 + w^2*x^93 + w*x^92 + w^2*x^91 + x^90 + w*x^88 + w*x^86 + w^2*x^85 + w^2*x^84 + w*x^82 + w*x^81 + x^80 + x^79 + w^2*x^78 + w*x^77 + w*x^76 + w^2*x^74 + w*x^73 + w^2*x^72 + w*x^71 + w^2*x^70 + w*x^69 + x^68 + w^2*x^67 + w*x^66 + w*x^65 + w*x^63 + w^2*x^62 + w*x^61 + w^2*x^60 + x^6 with shift constant w [3]: [105, 63, 18] Constacyclic by w Linear Code over GF(2^2) ConstaCyclicCode generated by x^103 + w^2*x^101 + w^2*x^100 + w^2*x^96 + x^95 + x^93 + x^90 + w^2*x^89 + w^2*x^88 + w^2*x^84 + w*x^82 + w^2*x^81 + x^80 + x^79 + w*x^78 + w*x^77 + w^2*x^76 + w*x^75 + w*x^74 + w^2*x^73 + x^72 + x^71 + x^70 + x^69 + w^2*x^68 + x^66 + w^2*x^65 + w*x^64 + x^63 + x^42 with shift constant w [4]: [109, 63, 20] Linear Code over GF(2^2) ConstructionX using [3] [2] and [1] last modified: 2020-09-23
Lb(109,63) = 19 is found by shortening of: Lb(110,64) = 19 Var Ub(109,63) = 33 follows by a one-step Griesmer bound from: Ub(75,62) = 8 is found by considering shortening to: Ub(67,54) = 8 LP
Var: From the Varshamov-Gilbert bound. Cf. R.R. Varshamov, Problems of the general theory of linear coding, Ph.D. thesis, Moscow State Univ., 1959. (Russian)
Notes
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