MinT - Best Known (223−22, 223, s)-Nets in Base 3

Best Known (223−22, 223, s)-Nets in Base 3

(223−22, 223, 762607)-Net over F₃ — Constructive and digital

Digital (201, 223, 762607)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
2. digital (189, 211, 762600)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²¹¹, 762600, F₃, 22, 22) (dual of [(762600, 22), 16776989, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(3²¹¹, 8388600, F₃, 22) (dual of [8388600, 8388389, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²¹¹, large, F₃, 22) (dual of [large, large−211, 23]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(223−22, 223, 2562000)-Net over F₃ — Digital

Digital (201, 223, 2562000)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²²³, 2562000, F₃, 3, 22) (dual of [(2562000, 3), 7685777, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²²³, 2796208, F₃, 3, 22) (dual of [(2796208, 3), 8388401, 23]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(3¹², 7, F₃, 3, 11) (dual of [(7, 3), 9, 12]-NRT-code), using
      - extended algebraic-geometric NRT-code AG_e(3;F,9P) [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
        an explicitly constructive algebraic function field [i]
    2. linear OOA(3²¹¹, 2796201, F₃, 3, 22) (dual of [(2796201, 3), 8388392, 23]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(3²¹¹, large, F₃, 22) (dual of [large, large−211, 23]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(223−22, 223, large)-Net in Base 3 — Upper bound on s

There is no (201, 223, large)-net in base 3, because

20 times m-reduction [i] would yield (201, 203, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]