Best Known (170−26, 170, s)-Nets in Base 2
(170−26, 170, 631)-Net over F2 — Constructive and digital
Digital (144, 170, 631)-net over F2, using
- net defined by OOA [i] based on linear OOA(2170, 631, F2, 26, 26) (dual of [(631, 26), 16236, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(2170, 8203, F2, 26) (dual of [8203, 8033, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(2170, 8205, F2, 26) (dual of [8205, 8035, 27]-code), using
- 1 times truncation [i] based on linear OA(2171, 8206, F2, 27) (dual of [8206, 8035, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2170, 8192, F2, 27) (dual of [8192, 8022, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2157, 8192, F2, 25) (dual of [8192, 8035, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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 X applied to Ce(26) ⊂ Ce(24) [i] based on
- 1 times truncation [i] based on linear OA(2171, 8206, F2, 27) (dual of [8206, 8035, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2170, 8205, F2, 26) (dual of [8205, 8035, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(2170, 8203, F2, 26) (dual of [8203, 8033, 27]-code), using
(170−26, 170, 2050)-Net over F2 — Digital
Digital (144, 170, 2050)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2170, 2050, F2, 4, 26) (dual of [(2050, 4), 8030, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2170, 2051, F2, 4, 26) (dual of [(2051, 4), 8034, 27]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2170, 8204, F2, 26) (dual of [8204, 8034, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(2170, 8205, F2, 26) (dual of [8205, 8035, 27]-code), using
- 1 times truncation [i] based on linear OA(2171, 8206, F2, 27) (dual of [8206, 8035, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2170, 8192, F2, 27) (dual of [8192, 8022, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2157, 8192, F2, 25) (dual of [8192, 8035, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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 X applied to Ce(26) ⊂ Ce(24) [i] based on
- 1 times truncation [i] based on linear OA(2171, 8206, F2, 27) (dual of [8206, 8035, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2170, 8205, F2, 26) (dual of [8205, 8035, 27]-code), using
- OOA 4-folding [i] based on linear OA(2170, 8204, F2, 26) (dual of [8204, 8034, 27]-code), using
- discarding factors / shortening the dual code based on linear OOA(2170, 2051, F2, 4, 26) (dual of [(2051, 4), 8034, 27]-NRT-code), using
(170−26, 170, 48953)-Net in Base 2 — Upper bound on s
There is no (144, 170, 48954)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1496 774984 926713 032456 728197 175319 328205 774519 042252 > 2170 [i]