MinT - Best Known (109−79, 109, s)-Nets in Base 3

Best Known (109−79, 109, s)-Nets in Base 3

(109−79, 109, 37)-Net over F₃ — Constructive and digital

Digital (30, 109, 37)-net over F₃, using

t-expansion [i] based on digital (27, 109, 37)-net over F₃, using
- net from sequence [i] based on digital (27, 36)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]

(109−79, 109, 42)-Net over F₃ — Digital

Digital (30, 109, 42)-net over F₃, using

t-expansion [i] based on digital (29, 109, 42)-net over F₃, using
- net from sequence [i] based on digital (29, 41)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 29 and N(F) ≥ 42, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(109−79, 109, 98)-Net over F₃ — Upper bound on s (digital)

There is no digital (30, 109, 99)-net over F₃, because

16 times m-reduction [i] would yield digital (30, 93, 99)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁹³, 99, F₃, 63) (dual of [99, 6, 64]-code), but
  - residual code [i] would yield linear OA(3³⁰, 35, F₃, 21) (dual of [35, 5, 22]-code), but
    - residual code [i] would yield linear OA(3⁹, 13, F₃, 7) (dual of [13, 4, 8]-code), but
      - 1 times truncation [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]

(109−79, 109, 99)-Net in Base 3 — Upper bound on s

There is no (30, 109, 100)-net in base 3, because

21 times m-reduction [i] would yield (30, 88, 100)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁸⁸, 100, S₃, 58), but
  - the linear programming bound shows that MÂ â‰¥ 19088 056323 407827 075424 486287 615602 692670 648963 / 19057 > 3⁸⁸ [i]