MinT - Best Known (138−121, 138, s)-Nets in Base 4

Best Known (138−121, 138, s)-Nets in Base 4

(138−121, 138, 33)-Net over F₄ — Constructive and digital

Digital (17, 138, 33)-net over F₄, using

t-expansion [i] based on digital (15, 138, 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]

(138−121, 138, 40)-Net over F₄ — Digital

Digital (17, 138, 40)-net over F₄, using

net from sequence [i] based on digital (17, 39)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 17 and N(F) ≥ 40, using
  - Drinfeld modules of rank 1 [i]

(138−121, 138, 65)-Net in Base 4 — Upper bound on s

There is no (17, 138, 66)-net in base 4, because

11 times m-reduction [i] would yield (17, 127, 66)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹²⁷, 66, S₄, 2, 110), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 1 100024 847754 503856 523924 357582 535124 606064 854323 585358 374847 048075 174731 579392 / 37 > 4¹²⁷ [i]