MinT - Best Known (72, 72+77, s)-Nets in Base 8

Best Known (72, 72+77, s)-Nets in Base 8

(72, 72+77, 130)-Net over F₈ — Constructive and digital

Digital (72, 149, 130)-net over F₈, using

11 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+77, 212)-Net over F₈ — Digital

Digital (72, 149, 212)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+77, 7039)-Net in Base 8 — Upper bound on s

There is no (72, 149, 7040)-net in base 8, because

1 times m-reduction [i] would yield (72, 148, 7040)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 45 593890 788653 700997 029527 645249 880095 580867 100957 752145 603408 904038 072302 457488 130394 990936 714177 283510 164866 798514 244739 552690 145645 > 8¹⁴⁸ [i]