MinT - Best Known (238−163, 238, s)-Nets in Base 4

Best Known (238−163, 238, s)-Nets in Base 4

(238−163, 238, 104)-Net over F₄ — Constructive and digital

Digital (75, 238, 104)-net over F₄, using

t-expansion [i] based on digital (73, 238, 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]

(238−163, 238, 112)-Net over F₄ — Digital

Digital (75, 238, 112)-net over F₄, using

t-expansion [i] based on digital (73, 238, 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]

(238−163, 238, 531)-Net in Base 4 — Upper bound on s

There is no (75, 238, 532)-net in base 4, because

1 times m-reduction [i] would yield (75, 237, 532)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 49602 456360 229888 806527 519182 568836 054841 581315 800857 939760 702391 502703 562358 411169 816489 594799 164385 814754 397284 171860 960433 833150 933062 303672 > 4²³⁷ [i]