MinT - Best Known (136−67, 136, s)-Nets in Base 4

Best Known (136−67, 136, s)-Nets in Base 4

(136−67, 136, 70)-Net over F₄ — Constructive and digital

Digital (69, 136, 70)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 36, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (33, 100, 56)-net over F₄, using
  - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
      - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(136−67, 136, 108)-Net over F₄ — Digital

Digital (69, 136, 108)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(136−67, 136, 1247)-Net in Base 4 — Upper bound on s

There is no (69, 136, 1248)-net in base 4, because

1 times m-reduction [i] would yield (69, 135, 1248)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1930 679033 708922 691074 683504 950197 369822 672539 174136 470462 279257 925704 675812 888430 > 4¹³⁵ [i]