Best Known (134, 134+18, s)-Nets in Base 2
(134, 134+18, 7286)-Net over F2 — Constructive and digital
Digital (134, 152, 7286)-net over F2, using
- t-expansion [i] based on digital (133, 152, 7286)-net over F2, using
- net defined by OOA [i] based on linear OOA(2152, 7286, F2, 19, 19) (dual of [(7286, 19), 138282, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [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
- OOA 9-folding and stacking with additional row [i] based on linear OA(2152, 65575, F2, 19) (dual of [65575, 65423, 20]-code), using
- net defined by OOA [i] based on linear OOA(2152, 7286, F2, 19, 19) (dual of [(7286, 19), 138282, 20]-NRT-code), using
(134, 134+18, 15135)-Net over F2 — Digital
Digital (134, 152, 15135)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2152, 15135, F2, 4, 18) (dual of [(15135, 4), 60388, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2152, 16393, F2, 4, 18) (dual of [(16393, 4), 65420, 19]-NRT-code), using
- strength reduction [i] based on linear OOA(2152, 16393, F2, 4, 19) (dual of [(16393, 4), 65420, 20]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2152, 65572, F2, 19) (dual of [65572, 65420, 20]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(2152, 65575, F2, 19) (dual of [65575, 65423, 20]-code), using
- OOA 4-folding [i] based on linear OA(2152, 65572, F2, 19) (dual of [65572, 65420, 20]-code), using
- strength reduction [i] based on linear OOA(2152, 16393, F2, 4, 19) (dual of [(16393, 4), 65420, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2152, 16393, F2, 4, 18) (dual of [(16393, 4), 65420, 19]-NRT-code), using
(134, 134+18, 503271)-Net in Base 2 — Upper bound on s
There is no (134, 152, 503272)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 5709 036861 570640 935184 932474 672719 109759 218188 > 2152 [i]