Best Known (230, 230+29, s)-Nets in Base 2
(230, 230+29, 18726)-Net over F2 — Constructive and digital
Digital (230, 259, 18726)-net over F2, using
- 23 times duplication [i] based on digital (227, 256, 18726)-net over F2, using
- net defined by OOA [i] based on linear OOA(2256, 18726, F2, 29, 29) (dual of [(18726, 29), 542798, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2256, 262165, F2, 29) (dual of [262165, 261909, 30]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2254, 262163, F2, 29) (dual of [262163, 261909, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(2253, 262144, F2, 29) (dual of [262144, 261891, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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(21, 19, F2, 1) (dual of [19, 18, 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(28) ⊂ Ce(26) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2254, 262163, F2, 29) (dual of [262163, 261909, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2256, 262165, F2, 29) (dual of [262165, 261909, 30]-code), using
- net defined by OOA [i] based on linear OOA(2256, 18726, F2, 29, 29) (dual of [(18726, 29), 542798, 30]-NRT-code), using
(230, 230+29, 35521)-Net over F2 — Digital
Digital (230, 259, 35521)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2259, 35521, F2, 7, 29) (dual of [(35521, 7), 248388, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2259, 37453, F2, 7, 29) (dual of [(37453, 7), 261912, 30]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2259, 262171, F2, 29) (dual of [262171, 261912, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2259, 262177, F2, 29) (dual of [262177, 261918, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2253, 262145, F2, 29) (dual of [262145, 261892, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 236−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2217, 262145, F2, 25) (dual of [262145, 261928, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 236−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2259, 262177, F2, 29) (dual of [262177, 261918, 30]-code), using
- OOA 7-folding [i] based on linear OA(2259, 262171, F2, 29) (dual of [262171, 261912, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(2259, 37453, F2, 7, 29) (dual of [(37453, 7), 261912, 30]-NRT-code), using
(230, 230+29, 2133076)-Net in Base 2 — Upper bound on s
There is no (230, 259, 2133077)-net in base 2, because
- 1 times m-reduction [i] would yield (230, 258, 2133077)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 463171 194176 746134 333302 652766 076448 112224 070497 758520 976621 061516 451687 110592 > 2258 [i]