MinT - Best Known (222−113, 222, s)-Nets in Base 4

Best Known (222−113, 222, s)-Nets in Base 4

(222−113, 222, 130)-Net over F₄ — Constructive and digital

Digital (109, 222, 130)-net over F₄, using

t-expansion [i] based on digital (105, 222, 130)-net over F₄, using
- net from sequence [i] based on digital (105, 129)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
    - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(222−113, 222, 165)-Net over F₄ — Digital

Digital (109, 222, 165)-net over F₄, using

net from sequence [i] based on digital (109, 164)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 109 and N(F) ≥ 165, using
  - Drinfeld modules of rank 1 [i]

(222−113, 222, 1674)-Net in Base 4 — Upper bound on s

There is no (109, 222, 1675)-net in base 4, because

1 times m-reduction [i] would yield (109, 221, 1675)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 652169 014514 947165 732602 204046 256272 725574 097956 028029 401210 458576 666747 858524 688740 275147 701221 772754 554326 671115 286088 106200 856280 > 4²²¹ [i]