MinT - Best Known (126−112, 126, s)-Nets in Base 4

Best Known (126−112, 126, s)-Nets in Base 4

(126−112, 126, 30)-Net over F₄ — Constructive and digital

Digital (14, 126, 30)-net over F₄, using

t-expansion [i] based on digital (13, 126, 30)-net over F₄, using
- net from sequence [i] based on digital (13, 29)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
    - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(126−112, 126, 33)-Net over F₄ — Digital

Digital (14, 126, 33)-net over F₄, using

t-expansion [i] based on digital (13, 126, 33)-net over F₄, using
- net from sequence [i] based on digital (13, 32)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 33, using
    - Drinfeld modules of rank 1 [i]

(126−112, 126, 55)-Net in Base 4 — Upper bound on s

There is no (14, 126, 56)-net in base 4, because

18 times m-reduction [i] would yield (14, 108, 56)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁰⁸, 56, S₄, 2, 94), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 13 479973 333575 319897 333507 543509 815336 818572 211270 286240 551805 124608 / 95 > 4¹⁰⁸ [i]