MinT - Best Known (222−123, 222, s)-Nets in Base 4

Best Known (222−123, 222, s)-Nets in Base 4

(222−123, 222, 104)-Net over F₄ — Constructive and digital

Digital (99, 222, 104)-net over F₄, using

t-expansion [i] based on digital (73, 222, 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]

(222−123, 222, 144)-Net over F₄ — Digital

Digital (99, 222, 144)-net over F₄, using

t-expansion [i] based on digital (91, 222, 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]

(222−123, 222, 1142)-Net in Base 4 — Upper bound on s

There is no (99, 222, 1143)-net in base 4, because

1 times m-reduction [i] would yield (99, 221, 1143)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 580513 244500 735833 473102 294907 000300 484235 839156 919251 367413 908479 712481 223297 815317 108357 319175 768047 490796 404151 657715 259869 927280 > 4²²¹ [i]