MinT - Best Known (253−177, 253, s)-Nets in Base 4

Best Known (253−177, 253, s)-Nets in Base 4

(253−177, 253, 104)-Net over F₄ — Constructive and digital

Digital (76, 253, 104)-net over F₄, using

t-expansion [i] based on digital (73, 253, 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]

(253−177, 253, 112)-Net over F₄ — Digital

Digital (76, 253, 112)-net over F₄, using

t-expansion [i] based on digital (73, 253, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(253−177, 253, 522)-Net in Base 4 — Upper bound on s

There is no (76, 253, 523)-net in base 4, because

1 times m-reduction [i] would yield (76, 252, 523)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52 846998 837219 056556 405422 293180 977259 297814 237767 899162 847494 804497 585021 961540 254151 426841 907869 347286 548510 618693 365781 836284 777484 848689 047979 872800 > 4²⁵² [i]