Best Known (230, 230+19, s)-Nets in Base 2
(230, 230+19, 932322)-Net over F2 — Constructive and digital
Digital (230, 249, 932322)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (32, 41, 256)-net over F2, using
- net defined by OOA [i] based on linear OOA(241, 256, F2, 9, 9) (dual of [(256, 9), 2263, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(241, 256, F2, 8, 9) (dual of [(256, 8), 2007, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(241, 1025, F2, 9) (dual of [1025, 984, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1025 | 220−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(241, 1025, F2, 9) (dual of [1025, 984, 10]-code), using
- appending kth column [i] based on linear OOA(241, 256, F2, 8, 9) (dual of [(256, 8), 2007, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(241, 256, F2, 9, 9) (dual of [(256, 9), 2263, 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 (32, 41, 256)-net over F2, using
(230, 230+19, 1678061)-Net over F2 — Digital
Digital (230, 249, 1678061)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2249, 1678061, F2, 5, 19) (dual of [(1678061, 5), 8390056, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(241, 341, F2, 5, 9) (dual of [(341, 5), 1664, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(241, 341, F2, 3, 9) (dual of [(341, 3), 982, 10]-NRT-code), using
- OOA 3-folding [i] based on linear OA(241, 1023, F2, 9) (dual of [1023, 982, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- OOA 3-folding [i] based on linear OA(241, 1023, F2, 9) (dual of [1023, 982, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(241, 341, F2, 3, 9) (dual of [(341, 3), 982, 10]-NRT-code), using
- linear OOA(2208, 1677720, F2, 5, 19) (dual of [(1677720, 5), 8388392, 20]-NRT-code), using
- OOA 5-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 5-folding [i] based on linear OA(2208, 8388600, F2, 19) (dual of [8388600, 8388392, 20]-code), using
- linear OOA(241, 341, F2, 5, 9) (dual of [(341, 5), 1664, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(230, 230+19, large)-Net in Base 2 — Upper bound on s
There is no (230, 249, large)-net in base 2, because
- 17 times m-reduction [i] would yield (230, 232, large)-net in base 2, but