MinT - Best Known (165−51, 165, s)-Nets in Base 4

Best Known (165−51, 165, s)-Nets in Base 4

(165−51, 165, 312)-Net over F₄ — Constructive and digital

Digital (114, 165, 312)-net over F₄, using

t-expansion [i] based on digital (113, 165, 312)-net over F₄, using
- trace code for nets [i] based on digital (3, 55, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(165−51, 165, 626)-Net over F₄ — Digital

Digital (114, 165, 626)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−51, 165, 30180)-Net in Base 4 — Upper bound on s

There is no (114, 165, 30181)-net in base 4, because

1 times m-reduction [i] would yield (114, 164, 30181)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 547 062203 328678 246451 345359 455323 889948 567587 529086 858578 700213 058192 472487 792060 108163 491572 145120 > 4¹⁶⁴ [i]