MinT - Best Known (109−73, 109, s)-Nets in Base 4

Best Known (109−73, 109, s)-Nets in Base 4

(109−73, 109, 56)-Net over F₄ — Constructive and digital

Digital (36, 109, 56)-net over F₄, using

t-expansion [i] based on digital (33, 109, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(109−73, 109, 65)-Net over F₄ — Digital

Digital (36, 109, 65)-net over F₄, using

t-expansion [i] based on digital (33, 109, 65)-net over F₄, using
- net from sequence [i] based on digital (33, 64)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(109−73, 109, 276)-Net in Base 4 — Upper bound on s

There is no (36, 109, 277)-net in base 4, because

1 times m-reduction [i] would yield (36, 108, 277)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 115909 724930 900650 674532 489069 813572 074707 358119 333545 566366 373064 > 4¹⁰⁸ [i]