Best Known (257−19, 257, s)-Nets in Base 2
(257−19, 257, 933090)-Net over F2 — Constructive and digital
Digital (238, 257, 933090)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (40, 49, 1024)-net over F2, using
- net defined by OOA [i] based on linear OOA(249, 1024, F2, 9, 9) (dual of [(1024, 9), 9167, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(249, 1024, F2, 8, 9) (dual of [(1024, 8), 8143, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(249, 4097, F2, 9) (dual of [4097, 4048, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, 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(249, 4097, F2, 9) (dual of [4097, 4048, 10]-code), using
- appending kth column [i] based on linear OOA(249, 1024, F2, 8, 9) (dual of [(1024, 8), 8143, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(249, 1024, F2, 9, 9) (dual of [(1024, 9), 9167, 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 (40, 49, 1024)-net over F2, using
(257−19, 257, 1679085)-Net over F2 — Digital
Digital (238, 257, 1679085)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2257, 1679085, F2, 5, 19) (dual of [(1679085, 5), 8395168, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(249, 1365, F2, 5, 9) (dual of [(1365, 5), 6776, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(249, 1365, F2, 3, 9) (dual of [(1365, 3), 4046, 10]-NRT-code), using
- OOA 3-folding [i] based on linear OA(249, 4095, F2, 9) (dual of [4095, 4046, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(249, 4096, F2, 9) (dual of [4096, 4047, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(249, 4096, F2, 9) (dual of [4096, 4047, 10]-code), using
- OOA 3-folding [i] based on linear OA(249, 4095, F2, 9) (dual of [4095, 4046, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(249, 1365, F2, 3, 9) (dual of [(1365, 3), 4046, 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(249, 1365, F2, 5, 9) (dual of [(1365, 5), 6776, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(257−19, 257, large)-Net in Base 2 — Upper bound on s
There is no (238, 257, large)-net in base 2, because
- 17 times m-reduction [i] would yield (238, 240, large)-net in base 2, but