MinT - Best Known (181−103, 181, s)-Nets in Base 4

Best Known (181−103, 181, s)-Nets in Base 4

(181−103, 181, 104)-Net over F₄ — Constructive and digital

Digital (78, 181, 104)-net over F₄, using

t-expansion [i] based on digital (73, 181, 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]

(181−103, 181, 112)-Net over F₄ — Digital

Digital (78, 181, 112)-net over F₄, using

t-expansion [i] based on digital (73, 181, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(181−103, 181, 841)-Net in Base 4 — Upper bound on s

There is no (78, 181, 842)-net in base 4, because

1 times m-reduction [i] would yield (78, 180, 842)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 471284 018666 423528 658470 746127 564700 008593 936491 749015 162367 770508 878266 986018 998523 292024 660702 825335 493520 > 4¹⁸⁰ [i]