MinT - Best Known (215−141, 215, s)-Nets in Base 4

Best Known (215−141, 215, s)-Nets in Base 4

(215−141, 215, 104)-Net over F₄ — Constructive and digital

Digital (74, 215, 104)-net over F₄, using

t-expansion [i] based on digital (73, 215, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(215−141, 215, 112)-Net over F₄ — Digital

Digital (74, 215, 112)-net over F₄, using

t-expansion [i] based on digital (73, 215, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(215−141, 215, 565)-Net in Base 4 — Upper bound on s

There is no (74, 215, 566)-net in base 4, because

1 times m-reduction [i] would yield (74, 214, 566)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 757 391973 740047 703227 540436 955206 025500 497254 679127 539171 591752 812444 429753 358501 205180 718356 891501 127274 226966 210693 239367 727872 > 4²¹⁴ [i]