MinT - Best Known (234−127, 234, s)-Nets in Base 4

Best Known (234−127, 234, s)-Nets in Base 4

(234−127, 234, 130)-Net over F₄ — Constructive and digital

Digital (107, 234, 130)-net over F₄, using

t-expansion [i] based on digital (105, 234, 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]

(234−127, 234, 144)-Net over F₄ — Digital

Digital (107, 234, 144)-net over F₄, using

t-expansion [i] based on digital (91, 234, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(234−127, 234, 1314)-Net in Base 4 — Upper bound on s

There is no (107, 234, 1315)-net in base 4, because

1 times m-reduction [i] would yield (107, 233, 1315)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 199 517479 895687 905902 215412 407251 974985 524136 247539 287886 940292 026351 298944 111559 621307 337882 058361 503090 317606 975275 691087 738572 025362 299904 > 4²³³ [i]