Best Known (186, 186+18, s)-Nets in Base 2
(186, 186+18, 466037)-Net over F2 — Constructive and digital
Digital (186, 204, 466037)-net over F2, using
- net defined by OOA [i] based on linear OOA(2204, 466037, F2, 18, 18) (dual of [(466037, 18), 8388462, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2204, 4194333, F2, 18) (dual of [4194333, 4194129, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2204, 4194335, F2, 18) (dual of [4194335, 4194131, 19]-code), using
- 1 times truncation [i] 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
- 1 times truncation [i] based on linear OA(2205, 4194336, F2, 19) (dual of [4194336, 4194131, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2204, 4194335, F2, 18) (dual of [4194335, 4194131, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(2204, 4194333, F2, 18) (dual of [4194333, 4194129, 19]-code), using
(186, 186+18, 699055)-Net over F2 — Digital
Digital (186, 204, 699055)-net over F2, using
- 22 times duplication [i] based on digital (184, 202, 699055)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2202, 699055, F2, 6, 18) (dual of [(699055, 6), 4194128, 19]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2202, 4194330, F2, 18) (dual of [4194330, 4194128, 19]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2199, 4194327, F2, 18) (dual of [4194327, 4194128, 19]-code), using
- 1 times truncation [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
- 1 times truncation [i] based on linear OA(2200, 4194328, F2, 19) (dual of [4194328, 4194128, 20]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2199, 4194327, F2, 18) (dual of [4194327, 4194128, 19]-code), using
- OOA 6-folding [i] based on linear OA(2202, 4194330, F2, 18) (dual of [4194330, 4194128, 19]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2202, 699055, F2, 6, 18) (dual of [(699055, 6), 4194128, 19]-NRT-code), using
(186, 186+18, large)-Net in Base 2 — Upper bound on s
There is no (186, 204, large)-net in base 2, because
- 16 times m-reduction [i] would yield (186, 188, large)-net in base 2, but