MinT - Best Known (254−163, 254, s)-Nets in Base 4

Best Known (254−163, 254, s)-Nets in Base 4

(254−163, 254, 104)-Net over F₄ — Constructive and digital

Digital (91, 254, 104)-net over F₄, using

t-expansion [i] based on digital (73, 254, 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]

(254−163, 254, 144)-Net over F₄ — Digital

Digital (91, 254, 144)-net over F₄, using

net from sequence [i] based on digital (91, 143)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
  - Drinfeld modules of rank 1 [i]

(254−163, 254, 719)-Net in Base 4 — Upper bound on s

There is no (91, 254, 720)-net in base 4, because

1 times m-reduction [i] would yield (91, 253, 720)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 231 263330 562088 836983 918430 769841 533344 797616 005524 067725 340310 934255 254676 740099 443323 381044 736676 817848 306791 137535 334137 430256 190776 404255 192333 717176 > 4²⁵³ [i]