MinT - Best Known (222−145, 222, s)-Nets in Base 4

Best Known (222−145, 222, s)-Nets in Base 4

(222−145, 222, 104)-Net over F₄ — Constructive and digital

Digital (77, 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]

(222−145, 222, 112)-Net over F₄ — Digital

Digital (77, 222, 112)-net over F₄, using

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

(222−145, 222, 591)-Net in Base 4 — Upper bound on s

There is no (77, 222, 592)-net in base 4, because

1 times m-reduction [i] would yield (77, 221, 592)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 807181 570114 683749 832361 448061 669726 275597 891820 182850 908442 022643 616498 032728 114503 356517 956838 811580 162012 577053 392933 721914 218875 > 4²²¹ [i]