MinT - Best Known (87, 87+166, s)-Nets in Base 4

Best Known (87, 87+166, s)-Nets in Base 4

(87, 87+166, 104)-Net over F₄ — Constructive and digital

Digital (87, 253, 104)-net over F₄, using

t-expansion [i] based on digital (73, 253, 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]

(87, 87+166, 129)-Net over F₄ — Digital

Digital (87, 253, 129)-net over F₄, using

t-expansion [i] based on digital (81, 253, 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]

(87, 87+166, 656)-Net in Base 4 — Upper bound on s

There is no (87, 253, 657)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 212 959326 967452 876481 534963 033953 543056 823131 804498 805522 302601 295792 545233 844900 704014 935145 986279 694832 002659 885182 789920 315499 678708 560768 880821 293568 > 4²⁵³ [i]