MinT - Best Known (167−151, 167, s)-Nets in Base 4

Best Known (167−151, 167, s)-Nets in Base 4

(167−151, 167, 33)-Net over F₄ — Constructive and digital

Digital (16, 167, 33)-net over F₄, using

t-expansion [i] based on digital (15, 167, 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]

(167−151, 167, 36)-Net over F₄ — Digital

Digital (16, 167, 36)-net over F₄, using

net from sequence [i] based on digital (16, 35)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 16 and N(F) ≥ 36, using
  - Drinfeld modules of rank 1 [i]

(167−151, 167, 62)-Net in Base 4 — Upper bound on s

There is no (16, 167, 63)-net in base 4, because

46 times m-reduction [i] would yield (16, 121, 63)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹²¹, 63, S₄, 2, 105), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 438 178072 065039 313736 657780 184243 791925 216006 425153 501614 158337 920569 704448 / 53 > 4¹²¹ [i]