MinT - Best Known (180−69, 180, s)-Nets in Base 4

Best Known (180−69, 180, s)-Nets in Base 4

(180−69, 180, 130)-Net over F₄ — Constructive and digital

Digital (111, 180, 130)-net over F₄, using

t-expansion [i] based on digital (105, 180, 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]

(180−69, 180, 308)-Net over F₄ — Digital

Digital (111, 180, 308)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(180−69, 180, 6640)-Net in Base 4 — Upper bound on s

There is no (111, 180, 6641)-net in base 4, because

1 times m-reduction [i] would yield (111, 179, 6641)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 588240 442724 338387 707391 719391 492997 282021 819246 480628 221333 908865 063881 118218 494075 285259 382320 845951 903112 > 4¹⁷⁹ [i]