MinT - Best Known (222−155, 222, s)-Nets in Base 4

Best Known (222−155, 222, s)-Nets in Base 4

(222−155, 222, 66)-Net over F₄ — Constructive and digital

Digital (67, 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]

(222−155, 222, 99)-Net over F₄ — Digital

Digital (67, 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]

(222−155, 222, 464)-Net in Base 4 — Upper bound on s

There is no (67, 222, 465)-net in base 4, because

1 times m-reduction [i] would yield (67, 221, 465)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 188670 100453 189684 508615 287080 678996 450758 515229 085579 583363 754293 478372 177105 634829 215749 068396 051225 197579 791139 461616 941517 602752 > 4²²¹ [i]