MinT - Best Known (253−147, 253, s)-Nets in Base 4

Best Known (253−147, 253, s)-Nets in Base 4

(253−147, 253, 130)-Net over F₄ — Constructive and digital

Digital (106, 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]

(253−147, 253, 144)-Net over F₄ — Digital

Digital (106, 253, 144)-net over F₄, using

t-expansion [i] based on digital (91, 253, 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]

(253−147, 253, 1058)-Net in Base 4 — Upper bound on s

There is no (106, 253, 1059)-net in base 4, because

1 times m-reduction [i] would yield (106, 252, 1059)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52 400449 986016 905905 423504 378405 484003 114123 432931 652537 025342 959498 870577 131916 267836 614331 364235 911156 501401 105964 040388 643700 794685 019964 187322 464384 > 4²⁵² [i]