Best Known (217, 217+19, s)-Nets in Base 2
(217, 217+19, 932112)-Net over F2 — Constructive and digital
Digital (217, 236, 932112)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (19, 28, 46)-net over F2, using
- net defined by OOA [i] based on linear OOA(228, 46, F2, 9, 9) (dual of [(46, 9), 386, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(228, 46, F2, 8, 9) (dual of [(46, 8), 340, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(29, 23, F2, 8, 4) (dual of [(23, 8), 175, 5]-NRT-code), using
- appending 4 arbitrary columns [i] based on linear OOA(29, 23, F2, 4, 4) (dual of [(23, 4), 83, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(29, 23, F2, 3, 4) (dual of [(23, 3), 60, 5]-NRT-code), using
- extracting embedded OOA [i] based on digital (5, 9, 23)-net over F2, using
- appending kth column [i] based on linear OOA(29, 23, F2, 3, 4) (dual of [(23, 3), 60, 5]-NRT-code), using
- appending 4 arbitrary columns [i] based on linear OOA(29, 23, F2, 4, 4) (dual of [(23, 4), 83, 5]-NRT-code), using
- linear OOA(219, 23, F2, 8, 9) (dual of [(23, 8), 165, 10]-NRT-code), using
- extracting embedded OOA [i] based on digital (10, 19, 23)-net over F2, using
- linear OOA(29, 23, F2, 8, 4) (dual of [(23, 8), 175, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(228, 46, F2, 8, 9) (dual of [(46, 8), 340, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(228, 46, F2, 9, 9) (dual of [(46, 9), 386, 10]-NRT-code), using
- digital (189, 208, 932066)-net over F2, using
- net defined by OOA [i] based on linear OOA(2208, 932066, F2, 19, 19) (dual of [(932066, 19), 17709046, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2208, 8388595, F2, 19) (dual of [8388595, 8388387, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2208, 8388595, F2, 19) (dual of [8388595, 8388387, 20]-code), using
- net defined by OOA [i] based on linear OOA(2208, 932066, F2, 19, 19) (dual of [(932066, 19), 17709046, 20]-NRT-code), using
- digital (19, 28, 46)-net over F2, using
(217, 217+19, 1398147)-Net over F2 — Digital
Digital (217, 236, 1398147)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2236, 1398147, F2, 6, 19) (dual of [(1398147, 6), 8388646, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(228, 47, F2, 6, 9) (dual of [(47, 6), 254, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(228, 47, F2, 2, 9) (dual of [(47, 2), 66, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(228, 94, F2, 9) (dual of [94, 66, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(228, 95, F2, 9) (dual of [95, 67, 10]-code), using
- adding a parity check bit [i] based on linear OA(227, 94, F2, 8) (dual of [94, 67, 9]-code), using
- a “Gra†code from Grassl’s database [i]
- adding a parity check bit [i] based on linear OA(227, 94, F2, 8) (dual of [94, 67, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(228, 95, F2, 9) (dual of [95, 67, 10]-code), using
- OOA 2-folding [i] based on linear OA(228, 94, F2, 9) (dual of [94, 66, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(228, 47, F2, 2, 9) (dual of [(47, 2), 66, 10]-NRT-code), using
- linear OOA(2208, 1398100, F2, 6, 19) (dual of [(1398100, 6), 8388392, 20]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2208, 8388600, F2, 19) (dual of [8388600, 8388392, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- OOA 6-folding [i] based on linear OA(2208, 8388600, F2, 19) (dual of [8388600, 8388392, 20]-code), using
- linear OOA(228, 47, F2, 6, 9) (dual of [(47, 6), 254, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(217, 217+19, large)-Net in Base 2 — Upper bound on s
There is no (217, 236, large)-net in base 2, because
- 17 times m-reduction [i] would yield (217, 219, large)-net in base 2, but