MinT - Best Known (104−35, 104, s)-Nets in Base 7

Best Known (104−35, 104, s)-Nets in Base 7

(104−35, 104, 200)-Net over F₇ — Constructive and digital

Digital (69, 104, 200)-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 (35, 70, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 35, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(104−35, 104, 885)-Net over F₇ — Digital

Digital (69, 104, 885)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(104−35, 104, 157784)-Net in Base 7 — Upper bound on s

There is no (69, 104, 157785)-net in base 7, because

1 times m-reduction [i] would yield (69, 103, 157785)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 1109 498608 633877 249646 253480 653070 734243 526122 543383 423080 673257 767047 241961 272484 886935 > 7¹⁰³ [i]