MinT - Best Known (109−37, 109, s)-Nets in Base 7

Best Known (109−37, 109, s)-Nets in Base 7

(109−37, 109, 192)-Net over F₇ — Constructive and digital

Digital (72, 109, 192)-net over F₇, using

1 times m-reduction [i] based on digital (72, 110, 192)-net over F₇, using
- trace code for nets [i] based on digital (17, 55, 96)-net over F₄₉, using
  - net from sequence [i] based on digital (17, 95)-sequence over F₄₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) ≤ 17 and N(F) ≥ 96, using
      - F₄ from the tower of function fields by GarcÃa, Stichtenoth, and RÃ¼ck over F₄₉ [i]

(109−37, 109, 880)-Net over F₇ — Digital

Digital (72, 109, 880)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−37, 109, 148092)-Net in Base 7 — Upper bound on s

There is no (72, 109, 148093)-net in base 7, because

1 times m-reduction [i] would yield (72, 108, 148093)-net in base 7, but
- 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]