MinT - Best Known (246−165, 246, s)-Nets in Base 4

Best Known (246−165, 246, s)-Nets in Base 4

(246−165, 246, 104)-Net over F₄ — Constructive and digital

Digital (81, 246, 104)-net over F₄, using

t-expansion [i] based on digital (73, 246, 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]

(246−165, 246, 129)-Net over F₄ — Digital

Digital (81, 246, 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]

(246−165, 246, 592)-Net in Base 4 — Upper bound on s

There is no (81, 246, 593)-net in base 4, because

1 times m-reduction [i] would yield (81, 245, 593)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3612 938692 286717 991471 721680 124560 646101 138247 587862 997653 136177 730608 090912 381625 320517 415353 995717 757307 898886 792712 890888 940803 790025 071729 946128 > 4²⁴⁵ [i]