MinT - Best Known (83−27, 83, s)-Nets in Base 7

Best Known (83−27, 83, s)-Nets in Base 7

(83−27, 83, 202)-Net over F₇ — Constructive and digital

Digital (56, 83, 202)-net over F₇, using

7¹ times duplication [i] based on digital (55, 82, 202)-net over F₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 26, 100)-net over F₇, using
    - trace code for nets [i] based on digital (0, 13, 50)-net over F₄₉, using
      - net from sequence [i] 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]
  2. digital (29, 56, 102)-net over F₇, using
    - trace code for nets [i] based on digital (1, 28, 51)-net over F₄₉, using
      - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
        Niederreiter sequence [i]

(83−27, 83, 890)-Net over F₇ — Digital

Digital (56, 83, 890)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(83−27, 83, 202234)-Net in Base 7 — Upper bound on s

There is no (56, 83, 202235)-net in base 7, because

1 times m-reduction [i] would yield (56, 82, 202235)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 1986 305626 716336 990259 859614 182240 368470 404122 572314 852929 861486 055643 > 7⁸² [i]