MinT - Best Known (231−135, 231, s)-Nets in Base 4

Best Known (231−135, 231, s)-Nets in Base 4

(231−135, 231, 104)-Net over F₄ — Constructive and digital

Digital (96, 231, 104)-net over F₄, using

t-expansion [i] based on digital (73, 231, 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]

(231−135, 231, 144)-Net over F₄ — Digital

Digital (96, 231, 144)-net over F₄, using

t-expansion [i] based on digital (91, 231, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(231−135, 231, 948)-Net in Base 4 — Upper bound on s

There is no (96, 231, 949)-net in base 4, because

1 times m-reduction [i] would yield (96, 230, 949)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 148943 019863 075522 866994 778872 571133 391168 611139 505639 374023 647665 238325 257738 193082 472203 506730 185168 053134 739956 317517 736546 251262 261400 > 4²³⁰ [i]