MinT - Best Known (223−139, 223, s)-Nets in Base 4

Best Known (223−139, 223, s)-Nets in Base 4

(223−139, 223, 104)-Net over F₄ — Constructive and digital

Digital (84, 223, 104)-net over F₄, using

t-expansion [i] based on digital (73, 223, 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]

(223−139, 223, 129)-Net over F₄ — Digital

Digital (84, 223, 129)-net over F₄, using

t-expansion [i] based on digital (81, 223, 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]

(223−139, 223, 709)-Net in Base 4 — Upper bound on s

There is no (84, 223, 710)-net in base 4, because

1 times m-reduction [i] would yield (84, 222, 710)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 47 637665 513362 394862 040339 499064 224176 772350 884160 529806 170209 611284 589476 129101 743703 669304 970687 968059 947760 218156 671108 605928 622465 > 4²²² [i]