MinT - Best Known (233−123, 233, s)-Nets in Base 4

Best Known (233−123, 233, s)-Nets in Base 4

(233−123, 233, 130)-Net over F₄ — Constructive and digital

Digital (110, 233, 130)-net over F₄, using

t-expansion [i] based on digital (105, 233, 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]

(233−123, 233, 165)-Net over F₄ — Digital

Digital (110, 233, 165)-net over F₄, using

t-expansion [i] based on digital (109, 233, 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]

(233−123, 233, 1481)-Net in Base 4 — Upper bound on s

There is no (110, 233, 1482)-net in base 4, because

1 times m-reduction [i] would yield (110, 232, 1482)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 49 510645 636898 779293 501054 347685 322788 981014 873087 896466 774360 408824 722893 128675 202081 294406 428776 423095 813414 956810 241736 995932 846403 769360 > 4²³² [i]