Best Known (133, 133+18, s)-Nets in Base 2
(133, 133+18, 7286)-Net over F2 — Constructive and digital
Digital (133, 151, 7286)-net over F2, using
- net defined by OOA [i] based on linear OOA(2151, 7286, F2, 18, 18) (dual of [(7286, 18), 130997, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2151, 65574, F2, 18) (dual of [65574, 65423, 19]-code), using
- 1 times truncation [i] based on linear OA(2152, 65575, F2, 19) (dual of [65575, 65423, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2145, 65536, F2, 19) (dual of [65536, 65391, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2113, 65536, F2, 15) (dual of [65536, 65423, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(18) ⊂ Ce(14) [i] based on
- 1 times truncation [i] based on linear OA(2152, 65575, F2, 19) (dual of [65575, 65423, 20]-code), using
- OA 9-folding and stacking [i] based on linear OA(2151, 65574, F2, 18) (dual of [65574, 65423, 19]-code), using
(133, 133+18, 14348)-Net over F2 — Digital
Digital (133, 151, 14348)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2151, 14348, F2, 4, 18) (dual of [(14348, 4), 57241, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2151, 16393, F2, 4, 18) (dual of [(16393, 4), 65421, 19]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2151, 65572, F2, 18) (dual of [65572, 65421, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2151, 65574, F2, 18) (dual of [65574, 65423, 19]-code), using
- 1 times truncation [i] based on linear OA(2152, 65575, F2, 19) (dual of [65575, 65423, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2145, 65536, F2, 19) (dual of [65536, 65391, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2113, 65536, F2, 15) (dual of [65536, 65423, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(18) ⊂ Ce(14) [i] based on
- 1 times truncation [i] based on linear OA(2152, 65575, F2, 19) (dual of [65575, 65423, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2151, 65574, F2, 18) (dual of [65574, 65423, 19]-code), using
- OOA 4-folding [i] based on linear OA(2151, 65572, F2, 18) (dual of [65572, 65421, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(2151, 16393, F2, 4, 18) (dual of [(16393, 4), 65421, 19]-NRT-code), using
(133, 133+18, 465965)-Net in Base 2 — Upper bound on s
There is no (133, 151, 465966)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2854 526447 276023 262300 803605 042680 517701 164645 > 2151 [i]