MinT - Best Known (42, 42+47, s)-Nets in Base 8

Best Known (42, 42+47, s)-Nets in Base 8

(42, 42+47, 98)-Net over F₈ — Constructive and digital

Digital (42, 89, 98)-net over F₈, using

t-expansion [i] based on digital (37, 89, 98)-net over F₈, using
- net from sequence [i] based on digital (37, 97)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 37 and N(F) ≥ 98, using
    - F₅ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]

(42, 42+47, 130)-Net over F₈ — Digital

Digital (42, 89, 130)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(8⁸⁹, 130, F₈, 3, 47) (dual of [(130, 3), 301, 48]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8⁸⁹, 131, F₈, 3, 47) (dual of [(131, 3), 304, 48]-NRT-code), using
  - constructionÂ X applied to AG(3;F,336P) âŠ‚ AG(3;F,341P) [i] based on
    1. linear OOA(8⁸⁵, 128, F₈, 3, 47) (dual of [(128, 3), 299, 48]-NRT-code), using algebraic-geometric NRT-code AG(3;F,336P) [i] based on function field F/F₈ with g(F) = 38 and N(F) ≥ 129, using
      - a curve from the manYPoints database [i]
    2. linear OOA(8⁸⁰, 128, F₈, 3, 42) (dual of [(128, 3), 304, 43]-NRT-code), using algebraic-geometric NRT-code AG(3;F,341P) [i] based on function field F/F₈ with g(F) = 38 and N(F) ≥ 129 (see above)
    3. linear OOA(8⁴, 3, F₈, 3, 4) (dual of [(3, 3), 5, 5]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁴, 8, F₈, 3, 4) (dual of [(8, 3), 20, 5]-NRT-code), using
        Reedâ€“Solomon NRT-code RS(3;20,8) [i]

(42, 42+47, 3828)-Net in Base 8 — Upper bound on s

There is no (42, 89, 3829)-net in base 8, because

1 times m-reduction [i] would yield (42, 88, 3829)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 29 708395 249201 437493 579442 395297 765303 395726 208042 891213 330970 548386 659216 198880 > 8⁸⁸ [i]