MinT - Best Known (225−123, 225, s)-Nets in Base 4

Best Known (225−123, 225, s)-Nets in Base 4

(225−123, 225, 104)-Net over F₄ — Constructive and digital

Digital (102, 225, 104)-net over F₄, using

t-expansion [i] based on digital (73, 225, 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]

(225−123, 225, 144)-Net over F₄ — Digital

Digital (102, 225, 144)-net over F₄, using

t-expansion [i] based on digital (91, 225, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(225−123, 225, 1226)-Net in Base 4 — Upper bound on s

There is no (102, 225, 1227)-net in base 4, because

1 times m-reduction [i] would yield (102, 224, 1227)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 737 974378 538900 140668 319110 983529 847585 636690 617643 009472 617162 238566 579309 687977 105275 918597 943562 930320 121988 058722 918223 992351 773792 > 4²²⁴ [i]