MinT - Best Known (253−133, 253, s)-Nets in Base 4

Best Known (253−133, 253, s)-Nets in Base 4

(253−133, 253, 130)-Net over F₄ — Constructive and digital

Digital (120, 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]

(253−133, 253, 168)-Net over F₄ — Digital

Digital (120, 253, 168)-net over F₄, using

t-expansion [i] based on digital (115, 253, 168)-net over F₄, using
- net from sequence [i] based on digital (115, 167)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 115 and N(F) ≥ 168, using
    - Drinfeld modules of rank 1 [i]

(253−133, 253, 1631)-Net in Base 4 — Upper bound on s

There is no (120, 253, 1632)-net in base 4, because

1 times m-reduction [i] would yield (120, 252, 1632)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52 964073 017891 632153 370643 406138 341346 993119 055479 219973 088602 307067 898208 541633 152924 146148 808451 149320 586610 222925 644764 696907 956779 527708 296650 689981 > 4²⁵² [i]