Best Known (128−15, 128, s)-Nets in Base 2
(128−15, 128, 37451)-Net over F2 — Constructive and digital
Digital (113, 128, 37451)-net over F2, using
- net defined by OOA [i] based on linear OOA(2128, 37451, F2, 15, 15) (dual of [(37451, 15), 561637, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2128, 262158, F2, 15) (dual of [262158, 262030, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2128, 262163, F2, 15) (dual of [262163, 262035, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2127, 262144, F2, 15) (dual of [262144, 262017, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2109, 262144, F2, 13) (dual of [262144, 262035, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(21, 19, F2, 1) (dual of [19, 18, 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(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(2128, 262163, F2, 15) (dual of [262163, 262035, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2128, 262158, F2, 15) (dual of [262158, 262030, 16]-code), using
(128−15, 128, 52432)-Net over F2 — Digital
Digital (113, 128, 52432)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2128, 52432, F2, 5, 15) (dual of [(52432, 5), 262032, 16]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2128, 262160, F2, 15) (dual of [262160, 262032, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2128, 262163, F2, 15) (dual of [262163, 262035, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2127, 262144, F2, 15) (dual of [262144, 262017, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2109, 262144, F2, 13) (dual of [262144, 262035, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(21, 19, F2, 1) (dual of [19, 18, 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(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(2128, 262163, F2, 15) (dual of [262163, 262035, 16]-code), using
- OOA 5-folding [i] based on linear OA(2128, 262160, F2, 15) (dual of [262160, 262032, 16]-code), using
(128−15, 128, 978269)-Net in Base 2 — Upper bound on s
There is no (113, 128, 978270)-net in base 2, because
- 1 times m-reduction [i] would yield (113, 127, 978270)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 170 142060 331023 460777 457650 503413 918360 > 2127 [i]