MinT - Best Known (82−37, 82, s)-Nets in Base 4

Best Known (82−37, 82, s)-Nets in Base 4

(82−37, 82, 90)-Net over F₄ — Constructive and digital

Digital (45, 82, 90)-net over F₄, using

trace code for nets [i] based on digital (4, 41, 45)-net over F₁₆, using
- net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
    - a function field by GarcÃa and Quoos [i]

(82−37, 82, 95)-Net over F₄ — Digital

Digital (45, 82, 95)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(82−37, 82, 1274)-Net in Base 4 — Upper bound on s

There is no (45, 82, 1275)-net in base 4, because

1 times m-reduction [i] would yield (45, 81, 1275)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5 875502 271134 535248 187859 541011 728774 750523 189321 > 4⁸¹ [i]