MinT - Best Known (258−151, 258, s)-Nets in Base 4

Best Known (258−151, 258, s)-Nets in Base 4

(258−151, 258, 130)-Net over F₄ — Constructive and digital

Digital (107, 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]

(258−151, 258, 144)-Net over F₄ — Digital

Digital (107, 258, 144)-net over F₄, using

t-expansion [i] based on digital (91, 258, 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]

(258−151, 258, 1047)-Net in Base 4 — Upper bound on s

There is no (107, 258, 1048)-net in base 4, because

1 times m-reduction [i] would yield (107, 257, 1048)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 56360 187176 330909 334452 058199 912374 072997 720687 843199 091856 265503 043632 268294 921153 314312 681456 219533 938381 095686 108365 226924 732762 947022 192407 028700 127968 > 4²⁵⁷ [i]