Best Known (64, 64+17, s)-Nets in Base 2
(64, 64+17, 152)-Net over F2 — Constructive and digital
Digital (64, 81, 152)-net over F2, using
- 21 times duplication [i] based on digital (63, 80, 152)-net over F2, using
- trace code for nets [i] based on digital (3, 20, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- trace code for nets [i] based on digital (3, 20, 38)-net over F16, using
(64, 64+17, 341)-Net over F2 — Digital
Digital (64, 81, 341)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(281, 341, F2, 3, 17) (dual of [(341, 3), 942, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(281, 1023, F2, 17) (dual of [1023, 942, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- OOA 3-folding [i] based on linear OA(281, 1023, F2, 17) (dual of [1023, 942, 18]-code), using
(64, 64+17, 3843)-Net in Base 2 — Upper bound on s
There is no (64, 81, 3844)-net in base 2, because
- 1 times m-reduction [i] would yield (64, 80, 3844)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1 210908 852356 486761 125280 > 280 [i]