MinT - Best Known (240−131, 240, s)-Nets in Base 4

Best Known (240−131, 240, s)-Nets in Base 4

(240−131, 240, 130)-Net over F₄ — Constructive and digital

Digital (109, 240, 130)-net over F₄, using

t-expansion [i] based on digital (105, 240, 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]

(240−131, 240, 165)-Net over F₄ — Digital

Digital (109, 240, 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]

(240−131, 240, 1312)-Net in Base 4 — Upper bound on s

There is no (109, 240, 1313)-net in base 4, because

1 times m-reduction [i] would yield (109, 239, 1313)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 791260 429021 039552 366381 490461 317010 792118 508151 163699 168415 121242 443497 100921 768006 561346 965815 513687 892498 386180 642931 201856 305942 307626 189200 > 4²³⁹ [i]