Best Known (245−27, 245, s)-Nets in Base 2
(245−27, 245, 20168)-Net over F2 — Constructive and digital
Digital (218, 245, 20168)-net over F2, using
- 23 times duplication [i] based on digital (215, 242, 20168)-net over F2, using
- net defined by OOA [i] based on linear OOA(2242, 20168, F2, 27, 27) (dual of [(20168, 27), 544294, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2242, 262185, F2, 27) (dual of [262185, 261943, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2242, 262187, F2, 27) (dual of [262187, 261945, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(2235, 262144, F2, 27) (dual of [262144, 261909, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2199, 262144, F2, 23) (dual of [262144, 261945, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(27, 43, F2, 3) (dual of [43, 36, 4]-code or 43-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(2242, 262187, F2, 27) (dual of [262187, 261945, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2242, 262185, F2, 27) (dual of [262185, 261943, 28]-code), using
- net defined by OOA [i] based on linear OOA(2242, 20168, F2, 27, 27) (dual of [(20168, 27), 544294, 28]-NRT-code), using
(245−27, 245, 37455)-Net over F2 — Digital
Digital (218, 245, 37455)-net over F2, using
- 23 times duplication [i] based on digital (215, 242, 37455)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2242, 37455, F2, 7, 27) (dual of [(37455, 7), 261943, 28]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2242, 262185, F2, 27) (dual of [262185, 261943, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2242, 262187, F2, 27) (dual of [262187, 261945, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(2235, 262144, F2, 27) (dual of [262144, 261909, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2199, 262144, F2, 23) (dual of [262144, 261945, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(27, 43, F2, 3) (dual of [43, 36, 4]-code or 43-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(2242, 262187, F2, 27) (dual of [262187, 261945, 28]-code), using
- OOA 7-folding [i] based on linear OA(2242, 262185, F2, 27) (dual of [262185, 261943, 28]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2242, 37455, F2, 7, 27) (dual of [(37455, 7), 261943, 28]-NRT-code), using
(245−27, 245, 2532237)-Net in Base 2 — Upper bound on s
There is no (218, 245, 2532238)-net in base 2, because
- 1 times m-reduction [i] would yield (218, 244, 2532238)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 28 269675 795206 389587 767094 395391 042306 835612 432513 853279 668881 847431 516099 > 2244 [i]