MinT - Best Known (72, 72+87, s)-Nets in Base 8

Best Known (72, 72+87, s)-Nets in Base 8

(72, 72+87, 130)-Net over F₈ — Constructive and digital

Digital (72, 159, 130)-net over F₈, using

1 times m-reduction [i] based on digital (72, 160, 130)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 58, 65)-net over F₈, using
    - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]
  2. digital (14, 102, 65)-net over F₈, using
    - net from sequence [i] based on digital (14, 64)-sequence over F₈ (see above)

(72, 72+87, 179)-Net over F₈ — Digital

Digital (72, 159, 179)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+87, 4992)-Net in Base 8 — Upper bound on s

There is no (72, 159, 4993)-net in base 8, because

1 times m-reduction [i] would yield (72, 158, 4993)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 48830 862052 781696 048413 580310 429586 538166 287152 991825 290411 855658 326506 032039 249925 260845 193811 897556 598404 088657 690622 424388 788093 132634 567040 > 8¹⁵⁸ [i]