MinT - Best Known (26, 124, s)-Nets in Base 3

Best Known (26, 124, s)-Nets in Base 3

(26, 124, 36)-Net over F₃ — Constructive and digital

Digital (26, 124, 36)-net over F₃, using

net from sequence [i] based on digital (26, 35)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using
  - an explicitly constructive algebraic function field [i]

(26, 124, 37)-Net over F₃ — Digital

Digital (26, 124, 37)-net over F₃, using

net from sequence [i] based on digital (26, 36)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 25, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 25 and N(F) ≥ 36, using algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(26, 124, 85)-Net over F₃ — Upper bound on s (digital)

There is no digital (26, 124, 86)-net over F₃, because

44 times m-reduction [i] would yield digital (26, 80, 86)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁸⁰, 86, F₃, 54) (dual of [86, 6, 55]-code), but
  - residual code [i] would yield linear OA(3²⁶, 31, F₃, 18) (dual of [31, 5, 19]-code), but
    - residual code [i] would yield linear OA(3⁸, 12, F₃, 6) (dual of [12, 4, 7]-code), but
      - residual code [i] would yield linear OA(3², 5, F₃, 2) (dual of [5, 3, 3]-code or 5-arc in PG(1,3)), but
        improvement by Maruta on the Griesmer bound [i]

(26, 124, 87)-Net in Base 3 — Upper bound on s

There is no (26, 124, 88)-net in base 3, because

44 times m-reduction [i] would yield (26, 80, 88)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁸⁰, 88, S₃, 54), but
  - the linear programming bound shows that MÂ â‰¥ 11972 515182 562019 788602 740026 717047 105681 / 70 > 3⁸⁰ [i]