Best Known (72, 72+19, s)-Nets in Base 2
(72, 72+19, 152)-Net over F2 — Constructive and digital
Digital (72, 91, 152)-net over F2, using
- 1 times m-reduction [i] based on digital (72, 92, 152)-net over F2, using
- trace code for nets [i] based on digital (3, 23, 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, 23, 38)-net over F16, using
(72, 72+19, 341)-Net over F2 — Digital
Digital (72, 91, 341)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(291, 341, F2, 3, 19) (dual of [(341, 3), 932, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(291, 1023, F2, 19) (dual of [1023, 932, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- OOA 3-folding [i] based on linear OA(291, 1023, F2, 19) (dual of [1023, 932, 20]-code), using
(72, 72+19, 4233)-Net in Base 2 — Upper bound on s
There is no (72, 91, 4234)-net in base 2, because
- 1 times m-reduction [i] would yield (72, 90, 4234)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1238 685394 290270 790534 217011 > 290 [i]