MinT - Best Known (111−101, 111, s)-Nets in Base 4

Best Known (111−101, 111, s)-Nets in Base 4

(111−101, 111, 27)-Net over F₄ — Constructive and digital

Digital (10, 111, 27)-net over F₄, using

net from sequence [i] based on digital (10, 26)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
  - an explicitly constructive algebraic function field [i]

(111−101, 111, 42)-Net in Base 4 — Upper bound on s

There is no (10, 111, 43)-net in base 4, because

30 times m-reduction [i] would yield (10, 81, 43)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4⁸¹, 43, S₄, 2, 71), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 20 461022 922632 640854 851587 658027 962275 183055 601664 / 3 > 4⁸¹ [i]