MinT - Best Known (258−141, 258, s)-Nets in Base 4

Best Known (258−141, 258, s)-Nets in Base 4

(258−141, 258, 130)-Net over F₄ — Constructive and digital

Digital (117, 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]

(258−141, 258, 168)-Net over F₄ — Digital

Digital (117, 258, 168)-net over F₄, using

t-expansion [i] based on digital (115, 258, 168)-net over F₄, using
- net from sequence [i] based on digital (115, 167)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 115 and N(F) ≥ 168, using
    - Drinfeld modules of rank 1 [i]

(258−141, 258, 1398)-Net in Base 4 — Upper bound on s

There is no (117, 258, 1399)-net in base 4, because

1 times m-reduction [i] would yield (117, 257, 1399)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 55060 961550 392591 634998 869422 889987 099152 195989 772701 308976 655041 764611 558520 555917 552625 669623 909395 344230 232454 889579 525076 422903 904450 198619 224576 529611 > 4²⁵⁷ [i]