MinT - Best Known (72, 72+33, s)-Nets in Base 4

Best Known (72, 72+33, s)-Nets in Base 4

(72, 72+33, 240)-Net over F₄ — Constructive and digital

Digital (72, 105, 240)-net over F₄, using

t-expansion [i] based on digital (71, 105, 240)-net over F₄, using
- trace code for nets [i] based on digital (1, 35, 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]

(72, 72+33, 412)-Net over F₄ — Digital

Digital (72, 105, 412)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+33, 18556)-Net in Base 4 — Upper bound on s

There is no (72, 105, 18557)-net in base 4, because

1 times m-reduction [i] would yield (72, 104, 18557)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 411 383897 018806 641178 105478 229060 483885 703167 890219 986103 459761 > 4¹⁰⁴ [i]