MinT - Best Known (109, 109+115, s)-Nets in Base 4

Best Known (109, 109+115, s)-Nets in Base 4

(109, 109+115, 130)-Net over F₄ — Constructive and digital

Digital (109, 224, 130)-net over F₄, using

t-expansion [i] based on digital (105, 224, 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]

(109, 109+115, 165)-Net over F₄ — Digital

Digital (109, 224, 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]

(109, 109+115, 1621)-Net in Base 4 — Upper bound on s

There is no (109, 224, 1622)-net in base 4, because

1 times m-reduction [i] would yield (109, 223, 1622)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 182 297955 221044 320009 316540 901456 680073 431266 898087 606464 755077 892900 694456 174588 956426 035799 440642 607883 333207 017938 139443 693650 666760 > 4²²³ [i]