MinT - Best Known (39−22, 39, s)-Nets in Base 7

Best Known (39−22, 39, s)-Nets in Base 7

(39−22, 39, 26)-Net over F₇ — Constructive and digital

Digital (17, 39, 26)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 13)-net over F₇, using
  - 1 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (5, 27, 13)-net over F₇, using
  - net from sequence [i] based on digital (5, 12)-sequence over F₇, using
    - Niederreiter sequence [i]

(39−22, 39, 50)-Net over F₇ — Digital

Digital (17, 39, 50)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(7³⁹, 50, F₇, 3, 22) (dual of [(50, 3), 111, 23]-NRT-code), using
- constructionÂ X applied to AG(3;F,118P) âŠ‚ AG(3;F,123P) [i] based on
  1. linear OOA(7³⁵, 47, F₇, 3, 22) (dual of [(47, 3), 106, 23]-NRT-code), using algebraic-geometric NRT-code AG(3;F,118P) [i] based on function field F/F₇ with g(F) = 13 and N(F) ≥ 48, using
    - an algebraic function field by Teo [i]
  2. linear OOA(7³⁰, 47, F₇, 3, 17) (dual of [(47, 3), 111, 18]-NRT-code), using algebraic-geometric NRT-code AG(3;F,123P) [i] based on function field F/F₇ with g(F) = 13 and N(F) ≥ 48 (see above)
  3. linear OOA(7⁴, 3, F₇, 3, 4) (dual of [(3, 3), 5, 5]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(7⁴, 7, F₇, 3, 4) (dual of [(7, 3), 17, 5]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(3;17,7) [i]

(39−22, 39, 804)-Net in Base 7 — Upper bound on s

There is no (17, 39, 805)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 917 339982 398151 929302 797205 503799 > 7³⁹ [i]