Best Known (251−30, 251, s)-Nets in Base 2
(251−30, 251, 4371)-Net over F2 — Constructive and digital
Digital (221, 251, 4371)-net over F2, using
- 24 times duplication [i] based on digital (217, 247, 4371)-net over F2, using
- t-expansion [i] based on digital (216, 247, 4371)-net over F2, using
- net defined by OOA [i] based on linear OOA(2247, 4371, F2, 31, 31) (dual of [(4371, 31), 135254, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2247, 65566, F2, 31) (dual of [65566, 65319, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2247, 65568, F2, 31) (dual of [65568, 65321, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- linear OA(2241, 65536, F2, 31) (dual of [65536, 65295, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2209, 65536, F2, 27) (dual of [65536, 65327, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(2247, 65568, F2, 31) (dual of [65568, 65321, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2247, 65566, F2, 31) (dual of [65566, 65319, 32]-code), using
- net defined by OOA [i] based on linear OOA(2247, 4371, F2, 31, 31) (dual of [(4371, 31), 135254, 32]-NRT-code), using
- t-expansion [i] based on digital (216, 247, 4371)-net over F2, using
(251−30, 251, 11903)-Net over F2 — Digital
Digital (221, 251, 11903)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2251, 11903, F2, 5, 30) (dual of [(11903, 5), 59264, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2251, 13115, F2, 5, 30) (dual of [(13115, 5), 65324, 31]-NRT-code), using
- 23 times duplication [i] based on linear OOA(2248, 13115, F2, 5, 30) (dual of [(13115, 5), 65327, 31]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2248, 65575, F2, 30) (dual of [65575, 65327, 31]-code), using
- strength reduction [i] based on linear OA(2248, 65575, F2, 31) (dual of [65575, 65327, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- linear OA(2241, 65536, F2, 31) (dual of [65536, 65295, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2209, 65536, F2, 27) (dual of [65536, 65327, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(27, 39, F2, 3) (dual of [39, 32, 4]-code or 39-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(30) ⊂ Ce(26) [i] based on
- strength reduction [i] based on linear OA(2248, 65575, F2, 31) (dual of [65575, 65327, 32]-code), using
- OOA 5-folding [i] based on linear OA(2248, 65575, F2, 30) (dual of [65575, 65327, 31]-code), using
- 23 times duplication [i] based on linear OOA(2248, 13115, F2, 5, 30) (dual of [(13115, 5), 65327, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2251, 13115, F2, 5, 30) (dual of [(13115, 5), 65324, 31]-NRT-code), using
(251−30, 251, 699823)-Net in Base 2 — Upper bound on s
There is no (221, 251, 699824)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 3618 575652 388184 310749 085102 421568 059469 670661 684790 747439 301332 151349 001771 > 2251 [i]