MinT - Best Known (111, 111+129, s)-Nets in Base 4

Best Known (111, 111+129, s)-Nets in Base 4

(111, 111+129, 130)-Net over F₄ — Constructive and digital

Digital (111, 240, 130)-net over F₄, using

t-expansion [i] based on digital (105, 240, 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+129, 165)-Net over F₄ — Digital

Digital (111, 240, 165)-net over F₄, using

t-expansion [i] based on digital (109, 240, 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+129, 1404)-Net in Base 4 — Upper bound on s

There is no (111, 240, 1405)-net in base 4, because

1 times m-reduction [i] would yield (111, 239, 1405)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 786965 098530 487562 598285 966857 906161 615165 862021 065180 904563 381591 038708 923321 215437 139210 064958 942316 208133 544529 942294 568831 805161 885010 792135 > 4²³⁹ [i]