MinT - Best Known (179−115, 179, s)-Nets in Base 4

Best Known (179−115, 179, s)-Nets in Base 4

(179−115, 179, 66)-Net over F₄ — Constructive and digital

Digital (64, 179, 66)-net over F₄, using

t-expansion [i] based on digital (49, 179, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(179−115, 179, 99)-Net over F₄ — Digital

Digital (64, 179, 99)-net over F₄, using

t-expansion [i] based on digital (61, 179, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(179−115, 179, 512)-Net in Base 4 — Upper bound on s

There is no (64, 179, 513)-net in base 4, because

1 times m-reduction [i] would yield (64, 178, 513)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 148182 660327 539709 635604 027468 889025 208956 179370 149700 291257 194606 351030 492255 053619 205940 607950 937735 489280 > 4¹⁷⁸ [i]