MinT - Best Known (224−22, 224, s)-Nets in Base 3

Best Known (224−22, 224, s)-Nets in Base 3

(224−22, 224, 762608)-Net over F₃ — Constructive and digital

Digital (202, 224, 762608)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 13, 8)-net over F₃, using
  - net from sequence [i] based on digital (2, 7)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 2 and N(F) ≥ 8, 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]

(224−22, 224, 2723241)-Net over F₃ — Digital

Digital (202, 224, 2723241)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²²⁴, 2723241, F₃, 3, 22) (dual of [(2723241, 3), 8169499, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²²⁴, 2796209, F₃, 3, 22) (dual of [(2796209, 3), 8388403, 23]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(3¹³, 8, F₃, 3, 11) (dual of [(8, 3), 11, 12]-NRT-code), using
      - extended algebraic-geometric NRT-code AG_e(3;F,12P) [i] based on function field F/F₃ with g(F) = 2 and N(F) ≥ 8, 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]

(224−22, 224, large)-Net in Base 3 — Upper bound on s

There is no (202, 224, large)-net in base 3, because

20 times m-reduction [i] would yield (202, 204, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]