Best Known (211−30, 211, s)-Nets in Base 2
(211−30, 211, 1093)-Net over F2 — Constructive and digital
Digital (181, 211, 1093)-net over F2, using
- net defined by OOA [i] based on linear OOA(2211, 1093, F2, 30, 30) (dual of [(1093, 30), 32579, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(2211, 16395, F2, 30) (dual of [16395, 16184, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2211, 16398, F2, 30) (dual of [16398, 16187, 31]-code), using
- 1 times truncation [i] based on linear OA(2212, 16399, F2, 31) (dual of [16399, 16187, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2211, 16384, F2, 31) (dual of [16384, 16173, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2197, 16384, F2, 29) (dual of [16384, 16187, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(21, 15, F2, 1) (dual of [15, 14, 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(30) ⊂ Ce(28) [i] based on
- 1 times truncation [i] based on linear OA(2212, 16399, F2, 31) (dual of [16399, 16187, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2211, 16398, F2, 30) (dual of [16398, 16187, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(2211, 16395, F2, 30) (dual of [16395, 16184, 31]-code), using
(211−30, 211, 3279)-Net over F2 — Digital
Digital (181, 211, 3279)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2211, 3279, F2, 5, 30) (dual of [(3279, 5), 16184, 31]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2211, 16395, F2, 30) (dual of [16395, 16184, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2211, 16398, F2, 30) (dual of [16398, 16187, 31]-code), using
- 1 times truncation [i] based on linear OA(2212, 16399, F2, 31) (dual of [16399, 16187, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2211, 16384, F2, 31) (dual of [16384, 16173, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2197, 16384, F2, 29) (dual of [16384, 16187, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(21, 15, F2, 1) (dual of [15, 14, 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(30) ⊂ Ce(28) [i] based on
- 1 times truncation [i] based on linear OA(2212, 16399, F2, 31) (dual of [16399, 16187, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2211, 16398, F2, 30) (dual of [16398, 16187, 31]-code), using
- OOA 5-folding [i] based on linear OA(2211, 16395, F2, 30) (dual of [16395, 16184, 31]-code), using
(211−30, 211, 110196)-Net in Base 2 — Upper bound on s
There is no (181, 211, 110197)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 3291 146901 242377 025597 269804 796919 992073 853534 764706 716643 621776 > 2211 [i]