MinT - Best Known (65−21, 65, s)-Nets in Base 7

Best Known (65−21, 65, s)-Nets in Base 7

(65−21, 65, 202)-Net over F₇ — Constructive and digital

Digital (44, 65, 202)-net over F₇, using

7¹ times duplication [i] based on digital (43, 64, 202)-net over F₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 20, 100)-net over F₇, using
    - trace code for nets [i] based on digital (0, 10, 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 (23, 44, 102)-net over F₇, using
    - trace code for nets [i] based on digital (1, 22, 51)-net over F₄₉, using
      - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
        Niederreiter sequence [i]

(65−21, 65, 782)-Net over F₇ — Digital

Digital (44, 65, 782)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(65−21, 65, 193392)-Net in Base 7 — Upper bound on s

There is no (44, 65, 193393)-net in base 7, because

1 times m-reduction [i] would yield (44, 64, 193393)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 1 219819 168739 813215 035572 180440 732229 309756 819524 267729 > 7⁶⁴ [i]