Best Known (187−19, 187, s)-Nets in Base 2
(187−19, 187, 116511)-Net over F2 — Constructive and digital
Digital (168, 187, 116511)-net over F2, using
- 23 times duplication [i] based on digital (165, 184, 116511)-net over F2, using
- net defined by OOA [i] based on linear OOA(2184, 116511, F2, 19, 19) (dual of [(116511, 19), 2213525, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2184, 1048600, F2, 19) (dual of [1048600, 1048416, 20]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2182, 1048598, F2, 19) (dual of [1048598, 1048416, 20]-code), using
- construction X4 applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(2181, 1048576, F2, 19) (dual of [1048576, 1048395, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2161, 1048576, F2, 17) (dual of [1048576, 1048415, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(221, 22, F2, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,2)), using
- dual of repetition code with length 22 [i]
- linear OA(21, 22, F2, 1) (dual of [22, 21, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(18) ⊂ Ce(16) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2182, 1048598, F2, 19) (dual of [1048598, 1048416, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2184, 1048600, F2, 19) (dual of [1048600, 1048416, 20]-code), using
- net defined by OOA [i] based on linear OOA(2184, 116511, F2, 19, 19) (dual of [(116511, 19), 2213525, 20]-NRT-code), using
(187−19, 187, 174768)-Net over F2 — Digital
Digital (168, 187, 174768)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2187, 174768, F2, 6, 19) (dual of [(174768, 6), 1048421, 20]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2187, 1048608, F2, 19) (dual of [1048608, 1048421, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2181, 1048576, F2, 19) (dual of [1048576, 1048395, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2141, 1048576, F2, 15) (dual of [1048576, 1048435, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(18) ⊂ Ce(14) [i] based on
- OOA 6-folding [i] based on linear OA(2187, 1048608, F2, 19) (dual of [1048608, 1048421, 20]-code), using
(187−19, 187, 6902989)-Net in Base 2 — Upper bound on s
There is no (168, 187, 6902990)-net in base 2, because
- 1 times m-reduction [i] would yield (168, 186, 6902990)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 98 079789 770697 557849 744133 453780 545212 885171 323100 760849 > 2186 [i]