MinT - Best Known (42, 42+35, s)-Nets in Base 4

Best Known (42, 42+35, s)-Nets in Base 4

(42, 42+35, 76)-Net over F₄ — Constructive and digital

Digital (42, 77, 76)-net over F₄, using

1 times m-reduction [i] based on digital (42, 78, 76)-net over F₄, using
- trace code for nets [i] based on digital (3, 39, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(42, 42+35, 89)-Net over F₄ — Digital

Digital (42, 77, 89)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 42+35, 1162)-Net in Base 4 — Upper bound on s

There is no (42, 77, 1163)-net in base 4, because

1 times m-reduction [i] would yield (42, 76, 1163)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5763 912773 141804 671257 010716 912040 273550 242000 > 4⁷⁶ [i]