MinT - Best Known (127, 127+129, s)-Nets in Base 4

Best Known (127, 127+129, s)-Nets in Base 4

(127, 127+129, 130)-Net over F₄ — Constructive and digital

Digital (127, 256, 130)-net over F₄, using

t-expansion [i] based on digital (105, 256, 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]

(127, 127+129, 176)-Net over F₄ — Digital

Digital (127, 256, 176)-net over F₄, using

t-expansion [i] based on digital (125, 256, 176)-net over F₄, using
- net from sequence [i] based on digital (125, 175)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 125 and N(F) ≥ 176, using
    - Drinfeld modules of rank 1 [i]

(127, 127+129, 2008)-Net in Base 4 — Upper bound on s

There is no (127, 256, 2009)-net in base 4, because

1 times m-reduction [i] would yield (127, 255, 2009)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3448 976256 498477 725601 185880 312504 977940 333333 966114 143195 299300 750352 848528 719995 783834 726824 152376 770221 665078 459864 053009 678794 065708 558579 723783 891135 > 4²⁵⁵ [i]