MinT - Best Known (72, 72+35, s)-Nets in Base 7

Best Known (72, 72+35, s)-Nets in Base 7

(72, 72+35, 202)-Net over F₇ — Constructive and digital

Digital (72, 107, 202)-net over F₇, using

7¹ times duplication [i] based on digital (71, 106, 202)-net over F₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (17, 34, 100)-net over F₇, using
    - trace code for nets [i] based on digital (0, 17, 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 (37, 72, 102)-net over F₇, using
    - trace code for nets [i] based on digital (1, 36, 51)-net over F₄₉, using
      - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
        Niederreiter sequence [i]

(72, 72+35, 1047)-Net over F₇ — Digital

Digital (72, 107, 1047)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+35, 222437)-Net in Base 7 — Upper bound on s

There is no (72, 107, 222438)-net in base 7, because

1 times m-reduction [i] would yield (72, 106, 222438)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 380552 856218 226130 736196 218533 199731 436003 894259 794406 635488 042907 474674 745009 793778 963909 > 7¹⁰⁶ [i]