Best Known (144−23, 144, s)-Nets in Base 2
(144−23, 144, 744)-Net over F2 — Constructive and digital
Digital (121, 144, 744)-net over F2, using
- net defined by OOA [i] based on linear OOA(2144, 744, F2, 23, 23) (dual of [(744, 23), 16968, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2144, 8185, F2, 23) (dual of [8185, 8041, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2144, 8192, F2, 23) (dual of [8192, 8048, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(2144, 8192, F2, 23) (dual of [8192, 8048, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2144, 8185, F2, 23) (dual of [8185, 8041, 24]-code), using
(144−23, 144, 1638)-Net over F2 — Digital
Digital (121, 144, 1638)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2144, 1638, F2, 5, 23) (dual of [(1638, 5), 8046, 24]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2144, 8190, F2, 23) (dual of [8190, 8046, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2144, 8192, F2, 23) (dual of [8192, 8048, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(2144, 8192, F2, 23) (dual of [8192, 8048, 24]-code), using
- OOA 5-folding [i] based on linear OA(2144, 8190, F2, 23) (dual of [8190, 8046, 24]-code), using
(144−23, 144, 40200)-Net in Base 2 — Upper bound on s
There is no (121, 144, 40201)-net in base 2, because
- 1 times m-reduction [i] would yield (121, 143, 40201)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 11 151937 449512 016429 291162 311856 117136 985128 > 2143 [i]