MinT - Best Known (72, 72+36, s)-Nets in Base 7

Best Known (72, 72+36, s)-Nets in Base 7

(72, 72+36, 200)-Net over F₇ — Constructive and digital

Digital (72, 108, 200)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (18, 36, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 18, 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 (36, 72, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 36, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(72, 72+36, 957)-Net over F₇ — Digital

Digital (72, 108, 957)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+36, 148092)-Net in Base 7 — Upper bound on s

There is no (72, 108, 148093)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 18 647046 281136 849795 021987 194317 601850 425066 903111 350701 085495 880769 938300 299188 240880 016329 > 7¹⁰⁸ [i]