Best Known (189, 189+27, s)-Nets in Base 2
(189, 189+27, 5044)-Net over F2 — Constructive and digital
Digital (189, 216, 5044)-net over F2, using
- net defined by OOA [i] based on linear OOA(2216, 5044, F2, 27, 27) (dual of [(5044, 27), 135972, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2216, 65573, F2, 27) (dual of [65573, 65357, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2216, 65575, F2, 27) (dual of [65575, 65359, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(2209, 65536, F2, 27) (dual of [65536, 65327, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2177, 65536, F2, 23) (dual of [65536, 65359, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(27, 39, F2, 3) (dual of [39, 32, 4]-code or 39-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2216, 65575, F2, 27) (dual of [65575, 65359, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2216, 65573, F2, 27) (dual of [65573, 65357, 28]-code), using
(189, 189+27, 10929)-Net over F2 — Digital
Digital (189, 216, 10929)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2216, 10929, F2, 6, 27) (dual of [(10929, 6), 65358, 28]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2216, 65574, F2, 27) (dual of [65574, 65358, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2216, 65575, F2, 27) (dual of [65575, 65359, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(2209, 65536, F2, 27) (dual of [65536, 65327, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2177, 65536, F2, 23) (dual of [65536, 65359, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(27, 39, F2, 3) (dual of [39, 32, 4]-code or 39-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2216, 65575, F2, 27) (dual of [65575, 65359, 28]-code), using
- OOA 6-folding [i] based on linear OA(2216, 65574, F2, 27) (dual of [65574, 65358, 28]-code), using
(189, 189+27, 539465)-Net in Base 2 — Upper bound on s
There is no (189, 216, 539466)-net in base 2, because
- 1 times m-reduction [i] would yield (189, 215, 539466)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 52656 445777 391741 362092 969655 898527 287603 412908 843386 233051 380672 > 2215 [i]