Best Known (209−15, 209, s)-Nets in Base 2
(209−15, 209, 1231137)-Net over F2 — Constructive and digital
Digital (194, 209, 1231137)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (40, 47, 32766)-net over F2, using
- net defined by OOA [i] based on linear OOA(247, 32766, F2, 7, 7) (dual of [(32766, 7), 229315, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(247, 32767, F2, 3, 7) (dual of [(32767, 3), 98254, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(247, 32766, F2, 7, 7) (dual of [(32766, 7), 229315, 8]-NRT-code), using
- digital (147, 162, 1198371)-net over F2, using
- net defined by OOA [i] based on linear OOA(2162, 1198371, F2, 15, 15) (dual of [(1198371, 15), 17975403, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2162, 8388598, F2, 15) (dual of [8388598, 8388436, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2162, large, F2, 15) (dual of [large, large−162, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(2162, large, F2, 15) (dual of [large, large−162, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2162, 8388598, F2, 15) (dual of [8388598, 8388436, 16]-code), using
- net defined by OOA [i] based on linear OOA(2162, 1198371, F2, 15, 15) (dual of [(1198371, 15), 17975403, 16]-NRT-code), using
- digital (40, 47, 32766)-net over F2, using
(209−15, 209, 2130501)-Net over F2 — Digital
Digital (194, 209, 2130501)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2209, 2130501, F2, 3, 15) (dual of [(2130501, 3), 6391294, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2209, 2828968, F2, 3, 15) (dual of [(2828968, 3), 8486695, 16]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(247, 32767, F2, 3, 7) (dual of [(32767, 3), 98254, 8]-NRT-code), using
- linear OOA(2162, 2796201, F2, 3, 15) (dual of [(2796201, 3), 8388441, 16]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2162, large, F2, 15) (dual of [large, large−162, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- OOA 3-folding [i] based on linear OA(2162, large, F2, 15) (dual of [large, large−162, 16]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(2209, 2828968, F2, 3, 15) (dual of [(2828968, 3), 8486695, 16]-NRT-code), using
(209−15, 209, large)-Net in Base 2 — Upper bound on s
There is no (194, 209, large)-net in base 2, because
- 13 times m-reduction [i] would yield (194, 196, large)-net in base 2, but