MinT - Best Known (249−159, 249, s)-Nets in Base 4

Best Known (249−159, 249, s)-Nets in Base 4

(249−159, 249, 104)-Net over F₄ — Constructive and digital

Digital (90, 249, 104)-net over F₄, using

t-expansion [i] based on digital (73, 249, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(249−159, 249, 129)-Net over F₄ — Digital

Digital (90, 249, 129)-net over F₄, using

t-expansion [i] based on digital (81, 249, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(249−159, 249, 718)-Net in Base 4 — Upper bound on s

There is no (90, 249, 719)-net in base 4, because

1 times m-reduction [i] would yield (90, 248, 719)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 208524 979137 364307 934074 661602 425715 281098 451170 594348 506173 843075 072858 514456 830649 576962 789245 499629 770946 920857 769425 205020 106049 327918 903709 655908 > 4²⁴⁸ [i]