Best Known (239−18, 239, s)-Nets in Base 2
(239−18, 239, 932125)-Net over F2 — Constructive and digital
Digital (221, 239, 932125)-net over F2, using
- 21 times duplication [i] based on digital (220, 238, 932125)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (22, 31, 58)-net over F2, using
- net defined by OOA [i] based on linear OOA(231, 58, F2, 9, 9) (dual of [(58, 9), 491, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(231, 58, F2, 8, 9) (dual of [(58, 8), 433, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(210, 32, F2, 8, 4) (dual of [(32, 8), 246, 5]-NRT-code), using
- appending 4 arbitrary columns [i] based on linear OOA(210, 32, F2, 4, 4) (dual of [(32, 4), 118, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(210, 32, F2, 3, 4) (dual of [(32, 3), 86, 5]-NRT-code), using
- extracting embedded OOA [i] based on digital (6, 10, 32)-net over F2, using
- appending kth column [i] based on linear OOA(210, 32, F2, 3, 4) (dual of [(32, 3), 86, 5]-NRT-code), using
- appending 4 arbitrary columns [i] based on linear OOA(210, 32, F2, 4, 4) (dual of [(32, 4), 118, 5]-NRT-code), using
- linear OOA(221, 29, F2, 8, 9) (dual of [(29, 8), 211, 10]-NRT-code), using
- extracting embedded OOA [i] based on digital (12, 21, 29)-net over F2, using
- linear OOA(210, 32, F2, 8, 4) (dual of [(32, 8), 246, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(231, 58, F2, 8, 9) (dual of [(58, 8), 433, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(231, 58, F2, 9, 9) (dual of [(58, 9), 491, 10]-NRT-code), using
- digital (189, 207, 932067)-net over F2, using
- net defined by OOA [i] based on linear OOA(2207, 932067, F2, 18, 18) (dual of [(932067, 18), 16776999, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- net defined by OOA [i] based on linear OOA(2207, 932067, F2, 18, 18) (dual of [(932067, 18), 16776999, 19]-NRT-code), using
- digital (22, 31, 58)-net over F2, using
- (u, u+v)-construction [i] based on
(239−18, 239, 1677806)-Net over F2 — Digital
Digital (221, 239, 1677806)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2239, 1677806, F2, 5, 18) (dual of [(1677806, 5), 8388791, 19]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(232, 86, F2, 5, 9) (dual of [(86, 5), 398, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(232, 86, F2, 2, 9) (dual of [(86, 2), 140, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(232, 172, F2, 9) (dual of [172, 140, 10]-code), using
- adding a parity check bit [i] based on linear OA(231, 171, F2, 8) (dual of [171, 140, 9]-code), using
- a “Gra†code from Grassl’s database [i]
- adding a parity check bit [i] based on linear OA(231, 171, F2, 8) (dual of [171, 140, 9]-code), using
- OOA 2-folding [i] based on linear OA(232, 172, F2, 9) (dual of [172, 140, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(232, 86, F2, 2, 9) (dual of [(86, 2), 140, 10]-NRT-code), using
- linear OOA(2207, 1677720, F2, 5, 18) (dual of [(1677720, 5), 8388393, 19]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2207, 8388600, F2, 18) (dual of [8388600, 8388393, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- OOA 5-folding [i] based on linear OA(2207, 8388600, F2, 18) (dual of [8388600, 8388393, 19]-code), using
- linear OOA(232, 86, F2, 5, 9) (dual of [(86, 5), 398, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(239−18, 239, large)-Net in Base 2 — Upper bound on s
There is no (221, 239, large)-net in base 2, because
- 16 times m-reduction [i] would yield (221, 223, large)-net in base 2, but