Best Known (56, 77, s)-Nets in Base 2
(56, 77, 75)-Net over F2 — Constructive and digital
Digital (56, 77, 75)-net over F2, using
- 1 times m-reduction [i] based on digital (56, 78, 75)-net over F2, using
- trace code for nets [i] based on digital (4, 26, 25)-net over F8, using
- net from sequence [i] based on digital (4, 24)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 4 and N(F) ≥ 25, using
- net from sequence [i] based on digital (4, 24)-sequence over F8, using
- trace code for nets [i] based on digital (4, 26, 25)-net over F8, using
(56, 77, 113)-Net over F2 — Digital
Digital (56, 77, 113)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(277, 113, F2, 2, 21) (dual of [(113, 2), 149, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(277, 128, F2, 2, 21) (dual of [(128, 2), 179, 22]-NRT-code), using
- OOA 2-folding [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]
- OOA 2-folding [i] based on linear OA(277, 256, F2, 21) (dual of [256, 179, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(277, 128, F2, 2, 21) (dual of [(128, 2), 179, 22]-NRT-code), using
(56, 77, 864)-Net in Base 2 — Upper bound on s
There is no (56, 77, 865)-net in base 2, because
- 1 times m-reduction [i] would yield (56, 76, 865)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 76220 426888 778947 527648 > 276 [i]