MinT - Best Known (239−137, 239, s)-Nets in Base 4

Best Known (239−137, 239, s)-Nets in Base 4

(239−137, 239, 104)-Net over F₄ — Constructive and digital

Digital (102, 239, 104)-net over F₄, using

t-expansion [i] based on digital (73, 239, 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]

(239−137, 239, 144)-Net over F₄ — Digital

Digital (102, 239, 144)-net over F₄, using

t-expansion [i] based on digital (91, 239, 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]

(239−137, 239, 1060)-Net in Base 4 — Upper bound on s

There is no (102, 239, 1061)-net in base 4, because

1 times m-reduction [i] would yield (102, 238, 1061)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 196791 369928 341407 372616 204867 130988 710055 255089 999614 832887 807930 546528 544759 500956 522737 398225 076831 355712 978760 127935 578023 305749 462584 781520 > 4²³⁸ [i]