MinT - Best Known (77−41, 77, s)-Nets in Base 7

Best Known (77−41, 77, s)-Nets in Base 7

(77−41, 77, 37)-Net over F₇ — Constructive and digital

Digital (36, 77, 37)-net over F₇, using

net from sequence [i] based on digital (36, 36)-sequence over F₇, using
- base reduction for sequences [i] based on digital (0, 36)-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]

(77−41, 77, 97)-Net over F₇ — Digital

Digital (36, 77, 97)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(7⁷⁷, 97, F₇, 3, 41) (dual of [(97, 3), 214, 42]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(7⁷⁴, 96, F₇, 3, 41) (dual of [(96, 3), 214, 42]-NRT-code), using
  - extended algebraic-geometric NRT-code AG_e(3;F,246P) [i] based on function field F/F₇ with g(F) = 33 and N(F) ≥ 96, using
    - an algebraic function field by Teo [i]

(77−41, 77, 2238)-Net in Base 7 — Upper bound on s

There is no (36, 77, 2239)-net in base 7, because

1 times m-reduction [i] would yield (36, 76, 2239)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 16886 269797 788388 214105 015629 619374 757645 592180 051718 642588 208009 > 7⁷⁶ [i]