MinT - Best Known (49, 49+51, s)-Nets in Base 8

Best Known (49, 49+51, s)-Nets in Base 8

(49, 49+51, 111)-Net over F₈ — Constructive and digital

Digital (49, 100, 111)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 35, 46)-net over F₈, using
  - net from sequence [i] based on digital (10, 45)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 9, N(F)Â =Â 45, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [i]
2. digital (14, 65, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]

(49, 49+51, 161)-Net over F₈ — Digital

Digital (49, 100, 161)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(49, 49+51, 5464)-Net in Base 8 — Upper bound on s

There is no (49, 100, 5465)-net in base 8, because

1 times m-reduction [i] would yield (49, 99, 5465)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 255519 652352 047647 640687 738904 729159 885656 136945 340776 801959 957944 927718 614283 412394 929064 > 8⁹⁹ [i]