MinT - Best Known (212−69, 212, s)-Nets in Base 4

Best Known (212−69, 212, s)-Nets in Base 4

(212−69, 212, 240)-Net over F₄ — Constructive and digital

Digital (143, 212, 240)-net over F₄, using

1 times m-reduction [i] based on digital (143, 213, 240)-net over F₄, using
- trace code for nets [i] based on digital (1, 71, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(212−69, 212, 632)-Net over F₄ — Digital

Digital (143, 212, 632)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−69, 212, 24557)-Net in Base 4 — Upper bound on s

There is no (143, 212, 24558)-net in base 4, because

1 times m-reduction [i] would yield (143, 211, 24558)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10 845223 941463 000657 736163 102719 026096 519534 263019 094227 906431 386119 457969 838789 803403 284514 640368 385485 164708 552001 713407 604410 > 4²¹¹ [i]