MinT - Best Known (230−35, 230, s)-Nets in Base 3

Best Known (230−35, 230, s)-Nets in Base 3

(230−35, 230, 1499)-Net over F₃ — Constructive and digital

Digital (195, 230, 1499)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (9, 26, 19)-net over F₃, using
  - net from sequence [i] based on digital (9, 18)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
      - an explicitly constructive algebraic function field [i]
2. digital (169, 204, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 51, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(230−35, 230, 13435)-Net over F₃ — Digital

Digital (195, 230, 13435)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²³⁰, 13435, F₃, 35) (dual of [13435, 13205, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3²³⁰, 19707, F₃, 35) (dual of [19707, 19477, 36]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(3²², 24, F₃, 17) (dual of [24, 2, 18]-code), using
      - repeating each code word 6 times [i] based on linear OA(3², 4, F₃, 2) (dual of [4, 2, 3]-code or 4-arc in PG(1,3)), using
        extended Reedâ€“Solomon code RS_e(2,3) [i]
        Hamming code H(2,3) [i]
        Simplex code S(2,3) [i]
        the Tetracode [i]
    2. linear OA(3²⁰⁸, 19683, F₃, 35) (dual of [19683, 19475, 36]-code), using
      - an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]

(230−35, 230, large)-Net in Base 3 — Upper bound on s

There is no (195, 230, large)-net in base 3, because

33 times m-reduction [i] would yield (195, 197, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]