MinT - Best Known (31, 31+37, s)-Nets in Base 81

Best Known (31, 31+37, s)-Nets in Base 81

(31, 31+37, 370)-Net over F₈₁ — Constructive and digital

Digital (31, 68, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 68, 370)-net over F₈₁, using
- net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
    - a function field by GarcÃa and Quoos [i]

(31, 31+37, 737)-Net over F₈₁ — Digital

Digital (31, 68, 737)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(31, 31+37, 1199063)-Net in Base 81 — Upper bound on s

There is no (31, 68, 1199064)-net in base 81, because

1 times m-reduction [i] would yield (31, 67, 1199064)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 73 875117 225893 343501 009339 916028 217899 170739 110183 803424 632536 599822 117929 050709 396697 220017 797988 474921 862617 976197 841666 108161 > 81⁶⁷ [i]