MinT - Best Known (258−147, 258, s)-Nets in Base 4

Best Known (258−147, 258, s)-Nets in Base 4

(258−147, 258, 130)-Net over F₄ — Constructive and digital

Digital (111, 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]

(258−147, 258, 165)-Net over F₄ — Digital

Digital (111, 258, 165)-net over F₄, using

t-expansion [i] based on digital (109, 258, 165)-net over F₄, using
- net from sequence [i] based on digital (109, 164)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 109 and N(F) ≥ 165, using
    - Drinfeld modules of rank 1 [i]

(258−147, 258, 1170)-Net in Base 4 — Upper bound on s

There is no (111, 258, 1171)-net in base 4, because

1 times m-reduction [i] would yield (111, 257, 1171)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 56053 019686 016110 636103 608087 187158 950139 284010 584703 354262 921777 019387 711787 884158 786530 975349 631156 060626 273208 442759 178710 484072 867810 808851 867571 403240 > 4²⁵⁷ [i]