MinT - Best Known (12, 28, s)-Nets in Base 7

Best Known (12, 28, s)-Nets in Base 7

(12, 28, 22)-Net over F₇ — Constructive and digital

Digital (12, 28, 22)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 13)-net over F₇, using
  - 4 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (3, 19, 11)-net over F₇, using
  - net from sequence [i] based on digital (3, 10)-sequence over F₇, using
    - Niederreiter sequence [i]
  - Niederreiter sequence (Bratleyâ€“Foxâ€“Niederreiter implementation) with equidistant coordinate [i]

(12, 28, 40)-Net over F₇ — Digital

Digital (12, 28, 40)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(7²⁸, 40, F₇, 2, 16) (dual of [(40, 2), 52, 17]-NRT-code), using
- constructionÂ X applied to AG(2;F,57P) âŠ‚ AG(2;F,61P) [i] based on
  1. linear OOA(7²⁵, 37, F₇, 2, 16) (dual of [(37, 2), 49, 17]-NRT-code), using algebraic-geometric NRT-code AG(2;F,57P) [i] based on function field F/F₇ with g(F) = 9 and N(F) ≥ 38, using
    - an algebraic function field over â„¤₇ by Yeo [i]
  2. linear OOA(7²¹, 37, F₇, 2, 12) (dual of [(37, 2), 53, 13]-NRT-code), using algebraic-geometric NRT-code AG(2;F,61P) [i] based on function field F/F₇ with g(F) = 9 and N(F) ≥ 38 (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]

(12, 28, 564)-Net in Base 7 — Upper bound on s

There is no (12, 28, 565)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 463005 562512 135393 884929 > 7²⁸ [i]