Best Known (223−14, 223, s)-Nets in Base 2
(223−14, 223, 2246945)-Net over F2 — Constructive and digital
Digital (209, 223, 2246945)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (55, 62, 1048574)-net over F2, using
- net defined by OOA [i] based on linear OOA(262, 1048574, F2, 7, 7) (dual of [(1048574, 7), 7339956, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(262, 1048575, F2, 3, 7) (dual of [(1048575, 3), 3145663, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(262, 1048574, F2, 7, 7) (dual of [(1048574, 7), 7339956, 8]-NRT-code), using
- digital (147, 161, 1198371)-net over F2, using
- net defined by OOA [i] based on linear OOA(2161, 1198371, F2, 14, 14) (dual of [(1198371, 14), 16777033, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(2161, 8388597, F2, 14) (dual of [8388597, 8388436, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(2161, large, F2, 14) (dual of [large, large−161, 15]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(2161, large, F2, 14) (dual of [large, large−161, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(2161, 8388597, F2, 14) (dual of [8388597, 8388436, 15]-code), using
- net defined by OOA [i] based on linear OOA(2161, 1198371, F2, 14, 14) (dual of [(1198371, 14), 16777033, 15]-NRT-code), using
- digital (55, 62, 1048574)-net over F2, using
(223−14, 223, 5242876)-Net over F2 — Digital
Digital (209, 223, 5242876)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2223, 5242876, F2, 2, 14) (dual of [(5242876, 2), 10485529, 15]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(262, 1048575, F2, 2, 7) (dual of [(1048575, 2), 2097088, 8]-NRT-code), using
- linear OOA(2161, 4194301, F2, 2, 14) (dual of [(4194301, 2), 8388441, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2161, 8388602, F2, 14) (dual of [8388602, 8388441, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(2161, large, F2, 14) (dual of [large, large−161, 15]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(2161, large, F2, 14) (dual of [large, large−161, 15]-code), using
- OOA 2-folding [i] based on linear OA(2161, 8388602, F2, 14) (dual of [8388602, 8388441, 15]-code), using
- (u, u+v)-construction [i] based on
(223−14, 223, large)-Net in Base 2 — Upper bound on s
There is no (209, 223, large)-net in base 2, because
- 12 times m-reduction [i] would yield (209, 211, large)-net in base 2, but