MinT - Best Known (111, 111+149, s)-Nets in Base 4

Best Known (111, 111+149, s)-Nets in Base 4

(111, 111+149, 130)-Net over F₄ — Constructive and digital

Digital (111, 260, 130)-net over F₄, using

t-expansion [i] based on digital (105, 260, 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]

(111, 111+149, 165)-Net over F₄ — Digital

Digital (111, 260, 165)-net over F₄, using

t-expansion [i] based on digital (109, 260, 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]

(111, 111+149, 1150)-Net in Base 4 — Upper bound on s

There is no (111, 260, 1151)-net in base 4, because

1 times m-reduction [i] would yield (111, 259, 1151)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 872113 878781 724435 026171 824775 164454 910854 972850 823469 650202 232707 971615 349699 658825 274622 161629 829570 824552 847332 566689 835728 660473 075776 162799 040177 499465 > 4²⁵⁹ [i]