Best Known (186, 186+19, s)-Nets in Base 2
(186, 186+19, 466037)-Net over F2 — Constructive and digital
Digital (186, 205, 466037)-net over F2, using
- net defined by OOA [i] based on linear OOA(2205, 466037, F2, 19, 19) (dual of [(466037, 19), 8854498, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2205, 4194334, F2, 19) (dual of [4194334, 4194129, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2205, 4194336, F2, 19) (dual of [4194336, 4194131, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2199, 4194304, F2, 19) (dual of [4194304, 4194105, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2155, 4194304, F2, 15) (dual of [4194304, 4194149, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2205, 4194336, F2, 19) (dual of [4194336, 4194131, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2205, 4194334, F2, 19) (dual of [4194334, 4194129, 20]-code), using
(186, 186+19, 599190)-Net over F2 — Digital
Digital (186, 205, 599190)-net over F2, using
- 23 times duplication [i] based on digital (183, 202, 599190)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2202, 599190, F2, 7, 19) (dual of [(599190, 7), 4194128, 20]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2202, 4194330, F2, 19) (dual of [4194330, 4194128, 20]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2200, 4194328, F2, 19) (dual of [4194328, 4194128, 20]-code), using
- construction X4 applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(2199, 4194304, F2, 19) (dual of [4194304, 4194105, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2177, 4194304, F2, 17) (dual of [4194304, 4194127, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(223, 24, F2, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,2)), using
- dual of repetition code with length 24 [i]
- linear OA(21, 24, F2, 1) (dual of [24, 23, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(18) ⊂ Ce(16) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2200, 4194328, F2, 19) (dual of [4194328, 4194128, 20]-code), using
- OOA 7-folding [i] based on linear OA(2202, 4194330, F2, 19) (dual of [4194330, 4194128, 20]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2202, 599190, F2, 7, 19) (dual of [(599190, 7), 4194128, 20]-NRT-code), using
(186, 186+19, large)-Net in Base 2 — Upper bound on s
There is no (186, 205, large)-net in base 2, because
- 17 times m-reduction [i] would yield (186, 188, large)-net in base 2, but