Best Known (256−26, 256, s)-Nets in Base 2
(256−26, 256, 40333)-Net over F2 — Constructive and digital
Digital (230, 256, 40333)-net over F2, using
- 21 times duplication [i] based on digital (229, 255, 40333)-net over F2, using
- t-expansion [i] based on digital (228, 255, 40333)-net over F2, using
- net defined by OOA [i] based on linear OOA(2255, 40333, F2, 27, 27) (dual of [(40333, 27), 1088736, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2255, 524330, F2, 27) (dual of [524330, 524075, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2255, 524333, F2, 27) (dual of [524333, 524078, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(2248, 524288, F2, 27) (dual of [524288, 524040, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2210, 524288, F2, 23) (dual of [524288, 524078, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(27, 45, F2, 3) (dual of [45, 38, 4]-code or 45-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2255, 524333, F2, 27) (dual of [524333, 524078, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2255, 524330, F2, 27) (dual of [524330, 524075, 28]-code), using
- net defined by OOA [i] based on linear OOA(2255, 40333, F2, 27, 27) (dual of [(40333, 27), 1088736, 28]-NRT-code), using
- t-expansion [i] based on digital (228, 255, 40333)-net over F2, using
(256−26, 256, 74904)-Net over F2 — Digital
Digital (230, 256, 74904)-net over F2, using
- 22 times duplication [i] based on digital (228, 254, 74904)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2254, 74904, F2, 7, 26) (dual of [(74904, 7), 524074, 27]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2254, 524328, F2, 26) (dual of [524328, 524074, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(2254, 524332, F2, 26) (dual of [524332, 524078, 27]-code), using
- 1 times truncation [i] based on linear OA(2255, 524333, F2, 27) (dual of [524333, 524078, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(2248, 524288, F2, 27) (dual of [524288, 524040, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2210, 524288, F2, 23) (dual of [524288, 524078, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(27, 45, F2, 3) (dual of [45, 38, 4]-code or 45-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- 1 times truncation [i] based on linear OA(2255, 524333, F2, 27) (dual of [524333, 524078, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2254, 524332, F2, 26) (dual of [524332, 524078, 27]-code), using
- OOA 7-folding [i] based on linear OA(2254, 524328, F2, 26) (dual of [524328, 524074, 27]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2254, 74904, F2, 7, 26) (dual of [(74904, 7), 524074, 27]-NRT-code), using
(256−26, 256, 4801531)-Net in Base 2 — Upper bound on s
There is no (230, 256, 4801532)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 115792 327345 107069 343773 701786 478613 354250 180996 535264 398818 359793 263812 742900 > 2256 [i]