MinT - Best Known (83, 83+139, s)-Nets in Base 4

Best Known (83, 83+139, s)-Nets in Base 4

(83, 83+139, 104)-Net over F₄ — Constructive and digital

Digital (83, 222, 104)-net over F₄, using

t-expansion [i] based on digital (73, 222, 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]

(83, 83+139, 129)-Net over F₄ — Digital

Digital (83, 222, 129)-net over F₄, using

t-expansion [i] based on digital (81, 222, 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]

(83, 83+139, 694)-Net in Base 4 — Upper bound on s

There is no (83, 222, 695)-net in base 4, because

1 times m-reduction [i] would yield (83, 221, 695)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 125044 918324 150073 041825 433188 056463 012655 354028 007135 525445 331880 610787 767261 489070 401225 457599 310511 863005 368462 458690 075722 269516 > 4²²¹ [i]