MinT - Best Known (251−231, 251, s)-Nets in Base 4

Best Known (251−231, 251, s)-Nets in Base 4

(251−231, 251, 33)-Net over F₄ — Constructive and digital

Digital (20, 251, 33)-net over F₄, using

t-expansion [i] based on digital (15, 251, 33)-net over F₄, using
- net from sequence [i] based on digital (15, 32)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
    - an explicitly constructive algebraic function field [i]

(251−231, 251, 41)-Net over F₄ — Digital

Digital (20, 251, 41)-net over F₄, using

t-expansion [i] based on digital (18, 251, 41)-net over F₄, using
- net from sequence [i] based on digital (18, 40)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 18 and N(F) ≥ 41, using
    - Drinfeld modules of rank 1 [i]

(251−231, 251, 72)-Net in Base 4 — Upper bound on s

There is no (20, 251, 73)-net in base 4, because

36 times m-reduction [i] would yield (20, 215, 73)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4²¹⁵, 73, S₄, 3, 195), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 169132 851341 369706 434283 265232 728105 909622 661597 924811 085399 254417 491899 776143 163005 438438 772352 514263 940863 868645 617881 567423 627264 / 49 > 4²¹⁵ [i]