Best Known (78−21, 78, s)-Nets in Base 2
(78−21, 78, 84)-Net over F2 — Constructive and digital
Digital (57, 78, 84)-net over F2, using
- trace code for nets [i] based on digital (5, 26, 28)-net over F8, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 5 and N(F) ≥ 28, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
(78−21, 78, 118)-Net over F2 — Digital
Digital (57, 78, 118)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(278, 118, F2, 2, 21) (dual of [(118, 2), 158, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(278, 132, F2, 2, 21) (dual of [(132, 2), 186, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(278, 264, F2, 21) (dual of [264, 186, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(278, 265, F2, 21) (dual of [265, 187, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(277, 256, F2, 21) (dual of [256, 179, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(269, 256, F2, 19) (dual of [256, 187, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 9, F2, 1) (dual of [9, 8, 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(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(278, 265, F2, 21) (dual of [265, 187, 22]-code), using
- OOA 2-folding [i] based on linear OA(278, 264, F2, 21) (dual of [264, 186, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(278, 132, F2, 2, 21) (dual of [(132, 2), 186, 22]-NRT-code), using
(78−21, 78, 927)-Net in Base 2 — Upper bound on s
There is no (57, 78, 928)-net in base 2, because
- 1 times m-reduction [i] would yield (57, 77, 928)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 152262 863183 333874 171669 > 277 [i]