MinT - Best Known (230−123, 230, s)-Nets in Base 4

Best Known (230−123, 230, s)-Nets in Base 4

(230−123, 230, 130)-Net over F₄ — Constructive and digital

Digital (107, 230, 130)-net over F₄, using

t-expansion [i] based on digital (105, 230, 130)-net over F₄, using
- net from sequence [i] based on digital (105, 129)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
    - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(230−123, 230, 144)-Net over F₄ — Digital

Digital (107, 230, 144)-net over F₄, using

t-expansion [i] based on digital (91, 230, 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]

(230−123, 230, 1380)-Net in Base 4 — Upper bound on s

There is no (107, 230, 1381)-net in base 4, because

1 times m-reduction [i] would yield (107, 229, 1381)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 770517 923571 437342 258461 996364 675675 069204 260608 345484 613353 066841 231815 475534 519188 117421 961594 829288 702490 889915 894355 295157 369407 068960 > 4²²⁹ [i]