MinT - Best Known (82, 82+155, s)-Nets in Base 4

Best Known (82, 82+155, s)-Nets in Base 4

(82, 82+155, 104)-Net over F₄ — Constructive and digital

Digital (82, 237, 104)-net over F₄, using

t-expansion [i] based on digital (73, 237, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(82, 82+155, 129)-Net over F₄ — Digital

Digital (82, 237, 129)-net over F₄, using

t-expansion [i] based on digital (81, 237, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(82, 82+155, 626)-Net in Base 4 — Upper bound on s

There is no (82, 237, 627)-net in base 4, because

1 times m-reduction [i] would yield (82, 236, 627)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12479 566425 340832 044082 689138 239942 630094 886927 527386 616153 855684 183447 721817 979355 737251 907844 016920 856215 545673 821361 398400 519104 566686 585560 > 4²³⁶ [i]