Best Known (219, 250, s)-Nets in Base 2
(219, 250, 4371)-Net over F2 — Constructive and digital
Digital (219, 250, 4371)-net over F2, using
- 23 times duplication [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
(219, 250, 10929)-Net over F2 — Digital
Digital (219, 250, 10929)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2250, 10929, F2, 6, 31) (dual of [(10929, 6), 65324, 32]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2248, 10929, F2, 6, 31) (dual of [(10929, 6), 65326, 32]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2248, 65574, F2, 31) (dual of [65574, 65326, 32]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(2248, 65575, F2, 31) (dual of [65575, 65327, 32]-code), using
- OOA 6-folding [i] based on linear OA(2248, 65574, F2, 31) (dual of [65574, 65326, 32]-code), using
- 22 times duplication [i] based on linear OOA(2248, 10929, F2, 6, 31) (dual of [(10929, 6), 65326, 32]-NRT-code), using
(219, 250, 638042)-Net in Base 2 — Upper bound on s
There is no (219, 250, 638043)-net in base 2, because
- 1 times m-reduction [i] would yield (219, 249, 638043)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 904 636750 746316 491053 620605 226877 655375 164379 541972 705995 400111 280738 845768 > 2249 [i]