MinT - Best Known (44, 94, s)-Nets in Base 8

Best Known (44, 94, s)-Nets in Base 8

(44, 94, 98)-Net over F₈ — Constructive and digital

Digital (44, 94, 98)-net over F₈, using

t-expansion [i] based on digital (37, 94, 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]

(44, 94, 131)-Net over F₈ — Digital

Digital (44, 94, 131)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(8⁹⁴, 131, F₈, 3, 50) (dual of [(131, 3), 299, 51]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8⁹⁴, 132, F₈, 3, 50) (dual of [(132, 3), 302, 51]-NRT-code), using
  - constructionÂ X applied to AG(3;F,333P) âŠ‚ AG(3;F,340P) [i] based on
    1. linear OOA(8⁸⁸, 128, F₈, 3, 50) (dual of [(128, 3), 296, 51]-NRT-code), using algebraic-geometric NRT-code AG(3;F,333P) [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(8⁸¹, 128, F₈, 3, 43) (dual of [(128, 3), 303, 44]-NRT-code), using algebraic-geometric NRT-code AG(3;F,340P) [i] based on function field F/F₈ with g(F) = 38 and N(F) ≥ 129 (see above)
    3. linear OOA(8⁶, 4, F₈, 3, 6) (dual of [(4, 3), 6, 7]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁶, 8, F₈, 3, 6) (dual of [(8, 3), 18, 7]-NRT-code), using
        Reedâ€“Solomon NRT-code RS(3;18,8) [i]

(44, 94, 3599)-Net in Base 8 — Upper bound on s

There is no (44, 94, 3600)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 7 783354 324569 293437 361012 651491 761066 707634 303569 259453 587000 718891 018235 521902 250784 > 8⁹⁴ [i]