MinT - Best Known (91−48, 91, s)-Nets in Base 8

Best Known (91−48, 91, s)-Nets in Base 8

(91−48, 91, 98)-Net over F₈ — Constructive and digital

Digital (43, 91, 98)-net over F₈, using

t-expansion [i] based on digital (37, 91, 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]

(91−48, 91, 131)-Net over F₈ — Digital

Digital (43, 91, 131)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(8⁹¹, 131, F₈, 3, 48) (dual of [(131, 3), 302, 49]-NRT-code), using
- strength reduction [i] based on linear OOA(8⁹¹, 131, F₈, 3, 49) (dual of [(131, 3), 302, 50]-NRT-code), using
  - constructionÂ X applied to AG(3;F,334P) âŠ‚ AG(3;F,339P) [i] based on
    1. linear OOA(8⁸⁷, 128, F₈, 3, 49) (dual of [(128, 3), 297, 50]-NRT-code), using algebraic-geometric NRT-code AG(3;F,334P) [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(8⁸², 128, F₈, 3, 44) (dual of [(128, 3), 302, 45]-NRT-code), using algebraic-geometric NRT-code AG(3;F,339P) [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]

(91−48, 91, 3704)-Net in Base 8 — Upper bound on s

There is no (43, 91, 3705)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 15220 390941 924704 149146 121508 020678 561482 770461 782556 428583 383059 706886 225835 398991 > 8⁹¹ [i]