MinT - Best Known (124−77, 124, s)-Nets in Base 4

Best Known (124−77, 124, s)-Nets in Base 4

(124−77, 124, 56)-Net over F₄ — Constructive and digital

Digital (47, 124, 56)-net over F₄, using

t-expansion [i] based on digital (33, 124, 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]

(124−77, 124, 81)-Net over F₄ — Digital

Digital (47, 124, 81)-net over F₄, using

t-expansion [i] based on digital (46, 124, 81)-net over F₄, using
- net from sequence [i] based on digital (46, 80)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 46 and N(F) ≥ 81, using
    - Drinfeld modules of rank 1 [i]

(124−77, 124, 414)-Net in Base 4 — Upper bound on s

There is no (47, 124, 415)-net in base 4, because

1 times m-reduction [i] would yield (47, 123, 415)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 115 638085 228841 092335 392270 700888 539276 838567 589735 521691 129600 580044 718836 > 4¹²³ [i]