MinT - Best Known (259−143, 259, s)-Nets in Base 4

Best Known (259−143, 259, s)-Nets in Base 4

(259−143, 259, 130)-Net over F₄ — Constructive and digital

Digital (116, 259, 130)-net over F₄, using

t-expansion [i] based on digital (105, 259, 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]

(259−143, 259, 168)-Net over F₄ — Digital

Digital (116, 259, 168)-net over F₄, using

t-expansion [i] based on digital (115, 259, 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]

(259−143, 259, 1342)-Net in Base 4 — Upper bound on s

There is no (116, 259, 1343)-net in base 4, because

1 times m-reduction [i] would yield (116, 258, 1343)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 217534 898007 451600 854913 665403 104437 392013 350490 647353 264315 618794 219301 722790 775115 844219 114630 155261 388191 896622 654250 574957 322875 426419 771384 337296 118000 > 4²⁵⁸ [i]