MinT - Best Known (81, 81+113, s)-Nets in Base 4

Best Known (81, 81+113, s)-Nets in Base 4

(81, 81+113, 104)-Net over F₄ — Constructive and digital

Digital (81, 194, 104)-net over F₄, using

t-expansion [i] based on digital (73, 194, 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]

(81, 81+113, 129)-Net over F₄ — Digital

Digital (81, 194, 129)-net over F₄, using

net from sequence [i] based on digital (81, 128)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
  - Drinfeld modules of rank 1 [i]

(81, 81+113, 814)-Net in Base 4 — Upper bound on s

There is no (81, 194, 815)-net in base 4, because

1 times m-reduction [i] would yield (81, 193, 815)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 160 612219 834748 398482 642351 053826 205394 672184 017445 555359 771763 473226 112213 967326 082763 537977 499438 223492 775471 855920 > 4¹⁹³ [i]