MinT - Best Known (253−173, 253, s)-Nets in Base 4

Best Known (253−173, 253, s)-Nets in Base 4

(253−173, 253, 104)-Net over F₄ — Constructive and digital

Digital (80, 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]

(253−173, 253, 112)-Net over F₄ — Digital

Digital (80, 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]

(253−173, 253, 567)-Net in Base 4 — Upper bound on s

There is no (80, 253, 568)-net in base 4, because

1 times m-reduction [i] would yield (80, 252, 568)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 57 938986 877335 572861 701778 193560 650356 052977 791651 171004 557104 695153 503327 988815 240974 951972 465650 505041 222931 393270 336490 379271 121951 693285 371793 680200 > 4²⁵² [i]