MinT - Best Known (16, 36, s)-Nets in Base 7

Best Known (16, 36, s)-Nets in Base 7

(16, 36, 26)-Net over F₇ — Constructive and digital

Digital (16, 36, 26)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 13)-net over F₇, using
  - 2 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (5, 25, 13)-net over F₇, using
  - net from sequence [i] based on digital (5, 12)-sequence over F₇, using
    - Niederreiter sequence [i]

(16, 36, 50)-Net over F₇ — Digital

Digital (16, 36, 50)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(7³⁶, 50, F₇, 2, 20) (dual of [(50, 2), 64, 21]-NRT-code), using
- constructionÂ X applied to AG(2;F,73P) âŠ‚ AG(2;F,77P) [i] based on
  1. linear OOA(7³³, 47, F₇, 2, 20) (dual of [(47, 2), 61, 21]-NRT-code), using algebraic-geometric NRT-code AG(2;F,73P) [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(7²⁹, 47, F₇, 2, 16) (dual of [(47, 2), 65, 17]-NRT-code), using algebraic-geometric NRT-code AG(2;F,77P) [i] based on function field F/F₇ with g(F) = 13 and N(F) ≥ 48 (see above)
  3. linear OOA(7³, 3, F₇, 2, 3) (dual of [(3, 2), 3, 4]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(7³, 7, F₇, 2, 3) (dual of [(7, 2), 11, 4]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(2;11,7) [i]

(16, 36, 825)-Net in Base 7 — Upper bound on s

There is no (16, 36, 826)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 2 653212 986958 592570 671237 567589 > 7³⁶ [i]