Best Known (186, 186+50, s)-Nets in Base 4
(186, 186+50, 1539)-Net over F4 — Constructive and digital
Digital (186, 236, 1539)-net over F4, using
- 1 times m-reduction [i] based on digital (186, 237, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 79, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 79, 513)-net over F64, using
(186, 186+50, 5082)-Net over F4 — Digital
Digital (186, 236, 5082)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4236, 5082, F4, 50) (dual of [5082, 4846, 51]-code), using
- 967 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 7 times 0, 1, 18 times 0, 1, 35 times 0, 1, 58 times 0, 1, 83 times 0, 1, 104 times 0, 1, 118 times 0, 1, 125 times 0, 1, 131 times 0, 1, 135 times 0, 1, 139 times 0) [i] based on linear OA(4223, 4102, F4, 50) (dual of [4102, 3879, 51]-code), using
- construction X applied to Ce(49) ⊂ Ce(48) [i] based on
- linear OA(4223, 4096, F4, 50) (dual of [4096, 3873, 51]-code), using an extension Ce(49) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,49], and designed minimum distance d ≥ |I|+1 = 50 [i]
- linear OA(4217, 4096, F4, 49) (dual of [4096, 3879, 50]-code), using an extension Ce(48) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,48], and designed minimum distance d ≥ |I|+1 = 49 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(49) ⊂ Ce(48) [i] based on
- 967 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 7 times 0, 1, 18 times 0, 1, 35 times 0, 1, 58 times 0, 1, 83 times 0, 1, 104 times 0, 1, 118 times 0, 1, 125 times 0, 1, 131 times 0, 1, 135 times 0, 1, 139 times 0) [i] based on linear OA(4223, 4102, F4, 50) (dual of [4102, 3879, 51]-code), using
(186, 186+50, 1636611)-Net in Base 4 — Upper bound on s
There is no (186, 236, 1636612)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 12194 496119 745129 262317 985554 956223 218068 171069 151869 473664 664993 131147 226494 136912 788442 595315 120901 268752 302752 036805 051656 743268 436603 357856 > 4236 [i]