MinT - Best Known (222−153, 222, s)-Nets in Base 4

Best Known (222−153, 222, s)-Nets in Base 4

(222−153, 222, 66)-Net over F₄ — Constructive and digital

Digital (69, 222, 66)-net over F₄, using

t-expansion [i] based on digital (49, 222, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(222−153, 222, 99)-Net over F₄ — Digital

Digital (69, 222, 99)-net over F₄, using

t-expansion [i] based on digital (61, 222, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(222−153, 222, 486)-Net in Base 4 — Upper bound on s

There is no (69, 222, 487)-net in base 4, because

1 times m-reduction [i] would yield (69, 221, 487)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 219296 369312 714272 872569 524429 245756 082941 819313 662458 050695 737777 173401 087725 123351 074346 388316 449535 244064 497160 792069 211747 587532 > 4²²¹ [i]