MinT - Best Known (258−159, 258, s)-Nets in Base 4

Best Known (258−159, 258, s)-Nets in Base 4

(258−159, 258, 104)-Net over F₄ — Constructive and digital

Digital (99, 258, 104)-net over F₄, using

t-expansion [i] based on digital (73, 258, 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]

(258−159, 258, 144)-Net over F₄ — Digital

Digital (99, 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]

(258−159, 258, 852)-Net in Base 4 — Upper bound on s

There is no (99, 258, 853)-net in base 4, because

1 times m-reduction [i] would yield (99, 257, 853)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 56746 269284 847483 230940 787495 227516 135012 691358 669917 828222 854670 848201 120418 393310 640653 902728 347071 435646 873106 934676 203611 833126 123964 691349 213764 955536 > 4²⁵⁷ [i]