MinT - Best Known (157−138, 157, s)-Nets in Base 4

Best Known (157−138, 157, s)-Nets in Base 4

(157−138, 157, 33)-Net over F₄ — Constructive and digital

Digital (19, 157, 33)-net over F₄, using

t-expansion [i] based on digital (15, 157, 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]

(157−138, 157, 41)-Net over F₄ — Digital

Digital (19, 157, 41)-net over F₄, using

t-expansion [i] based on digital (18, 157, 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]

(157−138, 157, 71)-Net in Base 4 — Upper bound on s

There is no (19, 157, 72)-net in base 4, because

17 times m-reduction [i] would yield (19, 140, 72)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁴⁰, 72, S₄, 2, 121), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 124 330809 102446 660538 845562 036705 210025 114037 699336 929360 115994 223289 874253 133343 883264 / 61 > 4¹⁴⁰ [i]