MinT - Best Known (109, 109+141, s)-Nets in Base 4

Best Known (109, 109+141, s)-Nets in Base 4

(109, 109+141, 130)-Net over F₄ — Constructive and digital

Digital (109, 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]

(109, 109+141, 165)-Net over F₄ — Digital

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]

(109, 109+141, 1185)-Net in Base 4 — Upper bound on s

There is no (109, 250, 1186)-net in base 4, because

1 times m-reduction [i] would yield (109, 249, 1186)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 849020 834591 575995 291961 757930 094068 278000 455811 496181 199256 388223 306211 791501 922656 367339 713637 241429 800667 528348 296126 821642 171620 011431 764337 815328 > 4²⁴⁹ [i]