MinT - Best Known (252−129, 252, s)-Nets in Base 4

Best Known (252−129, 252, s)-Nets in Base 4

(252−129, 252, 130)-Net over F₄ — Constructive and digital

Digital (123, 252, 130)-net over F₄, using

t-expansion [i] based on digital (105, 252, 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]

(252−129, 252, 168)-Net over F₄ — Digital

Digital (123, 252, 168)-net over F₄, using

t-expansion [i] based on digital (115, 252, 168)-net over F₄, using
- net from sequence [i] based on digital (115, 167)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 115 and N(F) ≥ 168, using
    - Drinfeld modules of rank 1 [i]

(252−129, 252, 1837)-Net in Base 4 — Upper bound on s

There is no (123, 252, 1838)-net in base 4, because

1 times m-reduction [i] would yield (123, 251, 1838)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 488982 571999 010516 204137 810289 553110 596148 308812 083847 188547 877293 103581 500872 933413 227611 869900 054600 331664 503850 098904 397797 685543 518933 580874 465740 > 4²⁵¹ [i]