MinT - Best Known (63−19, 63, s)-Nets in Base 7

Best Known (63−19, 63, s)-Nets in Base 7

(63−19, 63, 213)-Net over F₇ — Constructive and digital

Digital (44, 63, 213)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 7, 13)-net over F₇, using
  - 6 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (9, 18, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 9, 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]
3. digital (19, 38, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 19, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(63−19, 63, 1151)-Net over F₇ — Digital

Digital (44, 63, 1151)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(63−19, 63, 458544)-Net in Base 7 — Upper bound on s

There is no (44, 63, 458545)-net in base 7, because

1 times m-reduction [i] would yield (44, 62, 458545)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 24893 316586 302783 373687 313859 792746 931691 893229 293415 > 7⁶² [i]