MinT - Best Known (258−133, 258, s)-Nets in Base 4

Best Known (258−133, 258, s)-Nets in Base 4

(258−133, 258, 130)-Net over F₄ — Constructive and digital

Digital (125, 258, 130)-net over F₄, using

t-expansion [i] based on digital (105, 258, 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]

(258−133, 258, 176)-Net over F₄ — Digital

Digital (125, 258, 176)-net over F₄, using

net from sequence [i] based on digital (125, 175)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 125 and N(F) ≥ 176, using
  - Drinfeld modules of rank 1 [i]

(258−133, 258, 1818)-Net in Base 4 — Upper bound on s

There is no (125, 258, 1819)-net in base 4, because

1 times m-reduction [i] would yield (125, 257, 1819)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 54972 450792 134582 003924 665884 151384 409303 844673 723652 664016 041556 209637 890390 253136 867379 333200 121593 621104 279516 804043 525196 671092 913245 200725 808193 864708 > 4²⁵⁷ [i]