MinT - Best Known (78, 78+83, s)-Nets in Base 8

Best Known (78, 78+83, s)-Nets in Base 8

(78, 78+83, 130)-Net over F₈ — Constructive and digital

Digital (78, 161, 130)-net over F₈, using

t-expansion [i] based on digital (76, 161, 130)-net over F₈, using
- 11 times m-reduction [i] based on digital (76, 172, 130)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (14, 62, 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, 110, 65)-net over F₈, using
      - net from sequence [i] based on digital (14, 64)-sequence over F₈ (see above)

(78, 78+83, 229)-Net over F₈ — Digital

Digital (78, 161, 229)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(78, 78+83, 7684)-Net in Base 8 — Upper bound on s

There is no (78, 161, 7685)-net in base 8, because

1 times m-reduction [i] would yield (78, 160, 7685)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3 131401 014543 033800 552808 164912 900055 368374 598349 115641 045990 685150 493613 595682 617212 981633 457244 484461 000959 634933 966851 451921 916995 475577 588992 > 8¹⁶⁰ [i]