MinT - Best Known (168−53, 168, s)-Nets in Base 4

Best Known (168−53, 168, s)-Nets in Base 4

(168−53, 168, 312)-Net over F₄ — Constructive and digital

Digital (115, 168, 312)-net over F₄, using

trace code for nets [i] based on digital (3, 56, 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]

(168−53, 168, 585)-Net over F₄ — Digital

Digital (115, 168, 585)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−53, 168, 25875)-Net in Base 4 — Upper bound on s

There is no (115, 168, 25876)-net in base 4, because

1 times m-reduction [i] would yield (115, 167, 25876)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 35001 708065 935342 561166 187658 200025 462046 170131 560945 516020 413530 559349 843979 978510 221448 032540 626016 > 4¹⁶⁷ [i]