MinT - Best Known (82−47, 82, s)-Nets in Base 4

Best Known (82−47, 82, s)-Nets in Base 4

(82−47, 82, 56)-Net over F₄ — Constructive and digital

Digital (35, 82, 56)-net over F₄, using

t-expansion [i] based on digital (33, 82, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(82−47, 82, 65)-Net over F₄ — Digital

Digital (35, 82, 65)-net over F₄, using

t-expansion [i] based on digital (33, 82, 65)-net over F₄, using
- net from sequence [i] based on digital (33, 64)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(82−47, 82, 396)-Net in Base 4 — Upper bound on s

There is no (35, 82, 397)-net in base 4, because

1 times m-reduction [i] would yield (35, 81, 397)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6 092957 327954 716821 639263 698796 536583 402766 751168 > 4⁸¹ [i]