Best Known (136, 136+19, s)-Nets in Base 2
(136, 136+19, 14565)-Net over F2 — Constructive and digital
Digital (136, 155, 14565)-net over F2, using
- net defined by OOA [i] based on linear OOA(2155, 14565, F2, 19, 19) (dual of [(14565, 19), 276580, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2155, 131086, F2, 19) (dual of [131086, 130931, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2155, 131090, F2, 19) (dual of [131090, 130935, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(2154, 131072, F2, 19) (dual of [131072, 130918, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2137, 131072, F2, 17) (dual of [131072, 130935, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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 X applied to Ce(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(2155, 131090, F2, 19) (dual of [131090, 130935, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2155, 131086, F2, 19) (dual of [131086, 130931, 20]-code), using
(136, 136+19, 21848)-Net over F2 — Digital
Digital (136, 155, 21848)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2155, 21848, F2, 6, 19) (dual of [(21848, 6), 130933, 20]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2155, 131088, F2, 19) (dual of [131088, 130933, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2155, 131090, F2, 19) (dual of [131090, 130935, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(2154, 131072, F2, 19) (dual of [131072, 130918, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2137, 131072, F2, 17) (dual of [131072, 130935, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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 X applied to Ce(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(2155, 131090, F2, 19) (dual of [131090, 130935, 20]-code), using
- OOA 6-folding [i] based on linear OA(2155, 131088, F2, 19) (dual of [131088, 130933, 20]-code), using
(136, 136+19, 587083)-Net in Base 2 — Upper bound on s
There is no (136, 155, 587084)-net in base 2, because
- 1 times m-reduction [i] would yield (136, 154, 587084)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 22836 298804 441593 859873 380324 500321 706566 428201 > 2154 [i]