MinT - Best Known (189−101, 189, s)-Nets in Base 4

Best Known (189−101, 189, s)-Nets in Base 4

(189−101, 189, 104)-Net over F₄ — Constructive and digital

Digital (88, 189, 104)-net over F₄, using

t-expansion [i] based on digital (73, 189, 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]

(189−101, 189, 129)-Net over F₄ — Digital

Digital (88, 189, 129)-net over F₄, using

t-expansion [i] based on digital (81, 189, 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]

(189−101, 189, 1151)-Net in Base 4 — Upper bound on s

There is no (88, 189, 1152)-net in base 4, because

1 times m-reduction [i] would yield (88, 188, 1152)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 157878 916155 852980 099648 970835 513741 874799 424048 863033 428552 656741 112263 021417 173176 892169 307517 900904 594660 186285 > 4¹⁸⁸ [i]