MinT - Best Known (82, 82+143, s)-Nets in Base 4

Best Known (82, 82+143, s)-Nets in Base 4

(82, 82+143, 104)-Net over F₄ — Constructive and digital

Digital (82, 225, 104)-net over F₄, using

t-expansion [i] based on digital (73, 225, 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+143, 129)-Net over F₄ — Digital

Digital (82, 225, 129)-net over F₄, using

t-expansion [i] based on digital (81, 225, 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+143, 664)-Net in Base 4 — Upper bound on s

There is no (82, 225, 665)-net in base 4, because

1 times m-reduction [i] would yield (82, 224, 665)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 791 999056 632984 778112 617290 983077 060505 478482 127279 799626 522763 599756 188993 478498 794092 621594 904836 409511 809516 860229 270119 773446 369680 > 4²²⁴ [i]