MinT - Best Known (85−46, 85, s)-Nets in Base 7

Best Known (85−46, 85, s)-Nets in Base 7

(85−46, 85, 40)-Net over F₇ — Constructive and digital

Digital (39, 85, 40)-net over F₇, using

net from sequence [i] based on digital (39, 39)-sequence over F₇, using
- base reduction for sequences [i] based on digital (0, 39)-sequence over F₄₉, using
  - s-reduction based on digital (0, 49)-sequence over F₄₉, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50, using
      - the rational function field F₄₉(x) [i]
    - Niederreiter sequence [i]

(85−46, 85, 99)-Net over F₇ — Digital

Digital (39, 85, 99)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(7⁸⁵, 99, F₇, 3, 46) (dual of [(99, 3), 212, 47]-NRT-code), using
- constructionÂ X applied to AG(3;F,238P) âŠ‚ AG(3;F,245P) [i] based on
  1. linear OOA(7⁷⁹, 95, F₇, 3, 46) (dual of [(95, 3), 206, 47]-NRT-code), using algebraic-geometric NRT-code AG(3;F,238P) [i] based on function field F/F₇ with g(F) = 33 and N(F) ≥ 96, using
    - an algebraic function field by Teo [i]
  2. linear OOA(7⁷², 95, F₇, 3, 39) (dual of [(95, 3), 213, 40]-NRT-code), using algebraic-geometric NRT-code AG(3;F,245P) [i] based on function field F/F₇ with g(F) = 33 and N(F) ≥ 96 (see above)
  3. linear OOA(7⁶, 4, F₇, 3, 6) (dual of [(4, 3), 6, 7]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(7⁶, 7, F₇, 3, 6) (dual of [(7, 3), 15, 7]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(3;15,7) [i]

(85−46, 85, 2072)-Net in Base 7 — Upper bound on s

There is no (39, 85, 2073)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 688375 978531 038474 567084 033281 950721 942660 307685 496657 644834 256067 969463 > 7⁸⁵ [i]