MinT - Best Known (239−216, 239, s)-Nets in Base 4

Best Known (239−216, 239, s)-Nets in Base 4

(239−216, 239, 34)-Net over F₄ — Constructive and digital

Digital (23, 239, 34)-net over F₄, using

t-expansion [i] based on digital (21, 239, 34)-net over F₄, using
- net from sequence [i] based on digital (21, 33)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
    - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(239−216, 239, 45)-Net over F₄ — Digital

Digital (23, 239, 45)-net over F₄, using

net from sequence [i] based on digital (23, 44)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 23 and N(F) ≥ 45, using
  - a curve from the manYPoints database [i]

(239−216, 239, 85)-Net in Base 4 — Upper bound on s

There is no (23, 239, 86)-net in base 4, because

72 times m-reduction [i] would yield (23, 167, 86)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁶⁷, 86, S₄, 2, 144), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 5 389385 785865 341381 653845 487617 819044 230209 477720 875443 353098 774194 124190 027262 274822 736468 184804 622336 / 145 > 4¹⁶⁷ [i]