Best Known (234, 234+23, s)-Nets in Base 2
(234, 234+23, 762600)-Net over F2 — Constructive and digital
Digital (234, 257, 762600)-net over F2, using
- 23 times duplication [i] based on digital (231, 254, 762600)-net over F2, using
- net defined by OOA [i] based on linear OOA(2254, 762600, F2, 23, 23) (dual of [(762600, 23), 17539546, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2254, 8388601, F2, 23) (dual of [8388601, 8388347, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2254, large, F2, 23) (dual of [large, large−254, 24]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- discarding factors / shortening the dual code based on linear OA(2254, large, F2, 23) (dual of [large, large−254, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2254, 8388601, F2, 23) (dual of [8388601, 8388347, 24]-code), using
- net defined by OOA [i] based on linear OOA(2254, 762600, F2, 23, 23) (dual of [(762600, 23), 17539546, 24]-NRT-code), using
(234, 234+23, 1048575)-Net over F2 — Digital
Digital (234, 257, 1048575)-net over F2, using
- 23 times duplication [i] based on digital (231, 254, 1048575)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2254, 1048575, F2, 8, 23) (dual of [(1048575, 8), 8388346, 24]-NRT-code), using
- OOA 8-folding [i] based on linear OA(2254, 8388600, F2, 23) (dual of [8388600, 8388346, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2254, large, F2, 23) (dual of [large, large−254, 24]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- discarding factors / shortening the dual code based on linear OA(2254, large, F2, 23) (dual of [large, large−254, 24]-code), using
- OOA 8-folding [i] based on linear OA(2254, 8388600, F2, 23) (dual of [8388600, 8388346, 24]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2254, 1048575, F2, 8, 23) (dual of [(1048575, 8), 8388346, 24]-NRT-code), using
(234, 234+23, large)-Net in Base 2 — Upper bound on s
There is no (234, 257, large)-net in base 2, because
- 21 times m-reduction [i] would yield (234, 236, large)-net in base 2, but