MinT - Best Known (18, 40, s)-Nets in Base 8

Best Known (18, 40, s)-Nets in Base 8

(18, 40, 65)-Net over F₈ — Constructive and digital

Digital (18, 40, 65)-net over F₈, using

t-expansion [i] based on digital (14, 40, 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]

(18, 40, 67)-Net over F₈ — Digital

Digital (18, 40, 67)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(8⁴⁰, 67, F₈, 3, 22) (dual of [(67, 3), 161, 23]-NRT-code), using
- constructionÂ X applied to AG(3;F,169P) âŠ‚ AG(3;F,174P) [i] based on
  1. linear OOA(8³⁶, 64, F₈, 3, 22) (dual of [(64, 3), 156, 23]-NRT-code), using algebraic-geometric NRT-code AG(3;F,169P) [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
    - the Suzuki function field over F₈ [i]
  2. linear OOA(8³¹, 64, F₈, 3, 17) (dual of [(64, 3), 161, 18]-NRT-code), using algebraic-geometric NRT-code AG(3;F,174P) [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65 (see above)
  3. linear OOA(8⁴, 3, F₈, 3, 4) (dual of [(3, 3), 5, 5]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(8⁴, 8, F₈, 3, 4) (dual of [(8, 3), 20, 5]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(3;20,8) [i]

(18, 40, 1342)-Net in Base 8 — Upper bound on s

There is no (18, 40, 1343)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 339802 396106 502807 112293 725097 048724 > 8⁴⁰ [i]