MinT - Best Known (250−139, 250, s)-Nets in Base 4

Best Known (250−139, 250, s)-Nets in Base 4

(250−139, 250, 130)-Net over F₄ — Constructive and digital

Digital (111, 250, 130)-net over F₄, using

t-expansion [i] based on digital (105, 250, 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]

(250−139, 250, 165)-Net over F₄ — Digital

Digital (111, 250, 165)-net over F₄, using

t-expansion [i] based on digital (109, 250, 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]

(250−139, 250, 1259)-Net in Base 4 — Upper bound on s

There is no (111, 250, 1260)-net in base 4, because

1 times m-reduction [i] would yield (111, 249, 1260)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 828904 870382 685403 844818 277657 734831 221965 498020 772724 217092 340890 948751 519267 041378 241951 443174 857993 833130 140531 994946 325946 040126 459032 066561 409658 > 4²⁴⁹ [i]