MinT - Best Known (258−143, 258, s)-Nets in Base 4

Best Known (258−143, 258, s)-Nets in Base 4

(258−143, 258, 130)-Net over F₄ — Constructive and digital

Digital (115, 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]

(258−143, 258, 168)-Net over F₄ — Digital

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]

(258−143, 258, 1315)-Net in Base 4 — Upper bound on s

There is no (115, 258, 1316)-net in base 4, because

1 times m-reduction [i] would yield (115, 257, 1316)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 54572 602067 962289 927323 470798 995366 615554 319449 896405 618405 257289 105374 299669 036281 695816 444699 443200 799412 521250 436583 681901 375508 735166 580780 203495 849888 > 4²⁵⁷ [i]