Best Known (237−15, 237, s)-Nets in Base 2
(237−15, 237, 2396756)-Net over F2 — Constructive and digital
Digital (222, 237, 2396756)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (58, 65, 2097150)-net over F2, using
- net defined by OOA [i] based on linear OOA(265, 2097150, F2, 7, 7) (dual of [(2097150, 7), 14679985, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(265, 2097151, F2, 3, 7) (dual of [(2097151, 3), 6291388, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(265, 2097150, F2, 7, 7) (dual of [(2097150, 7), 14679985, 8]-NRT-code), using
- digital (157, 172, 1198378)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (3, 10, 7)-net over F2, using
- net from sequence [i] based on digital (3, 6)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 3 and N(F) ≥ 7, using
- Niederreiter–Xing sequence (Piršić implementation) with equidistant coordinate [i]
- net from sequence [i] based on digital (3, 6)-sequence over F2, 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 (3, 10, 7)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (58, 65, 2097150)-net over F2, using
(237−15, 237, 5592408)-Net over F2 — Digital
Digital (222, 237, 5592408)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2237, 5592408, F2, 3, 15) (dual of [(5592408, 3), 16776987, 16]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(268, 4194303, F2, 3, 7) (dual of [(4194303, 3), 12582841, 8]-NRT-code), using
- linear OOA(2169, 2796204, F2, 3, 15) (dual of [(2796204, 3), 8388443, 16]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(27, 3, F2, 3, 7) (dual of [(3, 3), 2, 8]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;2,2) [i]
- 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
- linear OOA(27, 3, F2, 3, 7) (dual of [(3, 3), 2, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- (u, u+v)-construction [i] based on
(237−15, 237, large)-Net in Base 2 — Upper bound on s
There is no (222, 237, large)-net in base 2, because
- 13 times m-reduction [i] would yield (222, 224, large)-net in base 2, but