MinT - Best Known (253−143, 253, s)-Nets in Base 4

Best Known (253−143, 253, s)-Nets in Base 4

(253−143, 253, 130)-Net over F₄ — Constructive and digital

Digital (110, 253, 130)-net over F₄, using

t-expansion [i] based on digital (105, 253, 130)-net over F₄, using
- net from sequence [i] based on digital (105, 129)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
    - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(253−143, 253, 165)-Net over F₄ — Digital

Digital (110, 253, 165)-net over F₄, using

t-expansion [i] based on digital (109, 253, 165)-net over F₄, using
- net from sequence [i] based on digital (109, 164)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 109 and N(F) ≥ 165, using
    - Drinfeld modules of rank 1 [i]

(253−143, 253, 1188)-Net in Base 4 — Upper bound on s

There is no (110, 253, 1189)-net in base 4, because

1 times m-reduction [i] would yield (110, 252, 1189)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 55 401261 107029 140988 062649 714559 027374 335579 821564 366123 907093 659385 608124 778067 986955 387167 134597 497212 412575 818834 723100 265275 353679 705474 783732 976704 > 4²⁵² [i]