MinT - Best Known (213−133, 213, s)-Nets in Base 4

Best Known (213−133, 213, s)-Nets in Base 4

(213−133, 213, 104)-Net over F₄ — Constructive and digital

Digital (80, 213, 104)-net over F₄, using

t-expansion [i] based on digital (73, 213, 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]

(213−133, 213, 112)-Net over F₄ — Digital

Digital (80, 213, 112)-net over F₄, using

t-expansion [i] based on digital (73, 213, 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]

(213−133, 213, 674)-Net in Base 4 — Upper bound on s

There is no (80, 213, 675)-net in base 4, because

1 times m-reduction [i] would yield (80, 212, 675)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 45 168466 147441 090238 053075 218637 037846 189797 112472 217013 143535 786388 825821 652960 030787 791512 284775 487357 947664 102264 002079 070570 > 4²¹² [i]