MinT - Best Known (58, 58+69, s)-Nets in Base 8

Best Known (58, 58+69, s)-Nets in Base 8

(58, 58+69, 111)-Net over F₈ — Constructive and digital

Digital (58, 127, 111)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 44, 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, 83, 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]

(58, 58+69, 152)-Net over F₈ — Digital

Digital (58, 127, 152)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(58, 58+69, 4275)-Net in Base 8 — Upper bound on s

There is no (58, 127, 4276)-net in base 8, because

1 times m-reduction [i] would yield (58, 126, 4276)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 619307 753167 571789 080588 277856 957888 410858 804712 291921 477403 813845 501339 838082 771505 743713 539499 012098 051876 698705 > 8¹²⁶ [i]