MinT - Best Known (81, 81+137, s)-Nets in Base 4

Best Known (81, 81+137, s)-Nets in Base 4

(81, 81+137, 104)-Net over F₄ — Constructive and digital

Digital (81, 218, 104)-net over F₄, using

t-expansion [i] based on digital (73, 218, 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]

(81, 81+137, 129)-Net over F₄ — Digital

Digital (81, 218, 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]

(81, 81+137, 672)-Net in Base 4 — Upper bound on s

There is no (81, 218, 673)-net in base 4, because

1 times m-reduction [i] would yield (81, 217, 673)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 45446 838533 882241 776005 297626 310249 065412 670122 639781 910896 974542 690961 858466 043888 348529 908677 033024 484469 890952 246956 296400 634000 > 4²¹⁷ [i]