MinT - Best Known (109, 109+145, s)-Nets in Base 4

Best Known (109, 109+145, s)-Nets in Base 4

(109, 109+145, 130)-Net over F₄ — Constructive and digital

Digital (109, 254, 130)-net over F₄, using

t-expansion [i] based on digital (105, 254, 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+145, 165)-Net over F₄ — Digital

Digital (109, 254, 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+145, 1143)-Net in Base 4 — Upper bound on s

There is no (109, 254, 1144)-net in base 4, because

1 times m-reduction [i] would yield (109, 253, 1144)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 213 493492 999197 977021 415151 493979 046966 307911 374601 050306 731119 900081 132640 885930 941719 230316 528416 234926 243405 264507 628100 667284 383052 323081 660826 324970 > 4²⁵³ [i]