MinT - Best Known (234−21, 234, s)-Nets in Base 3

Best Known (234−21, 234, s)-Nets in Base 3

(234−21, 234, 838900)-Net over F₃ — Constructive and digital

Digital (213, 234, 838900)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (14, 24, 40)-net over F₃, using
  - trace code for nets [i] based on digital (2, 12, 20)-net over F₉, using
    - net from sequence [i] based on digital (2, 19)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 2 and N(F) ≥ 20, using
        a function field by GarcÃa and Quoos [i]
2. digital (189, 210, 838860)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²¹⁰, 838860, F₃, 21, 21) (dual of [(838860, 21), 17615850, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(3²¹⁰, 8388601, F₃, 21) (dual of [8388601, 8388391, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²¹⁰, large, F₃, 21) (dual of [large, large−210, 22]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(234−21, 234, 4194341)-Net over F₃ — Digital

Digital (213, 234, 4194341)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²³⁴, 4194341, F₃, 2, 21) (dual of [(4194341, 2), 8388448, 22]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3²⁴, 40, F₃, 2, 10) (dual of [(40, 2), 56, 11]-NRT-code), using
    - extracting embedded OOA [i] based on digital (14, 24, 40)-net over F₃, using
      - trace code for nets [i] based on digital (2, 12, 20)-net over F₉, using
        net from sequence [i] based on digital (2, 19)-sequence over F₉, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 2 and N(F) ≥ 20, using
        a function field by GarcÃa and Quoos [i]
  2. linear OOA(3²¹⁰, 4194301, F₃, 2, 21) (dual of [(4194301, 2), 8388392, 22]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3²¹⁰, 8388602, F₃, 21) (dual of [8388602, 8388392, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²¹⁰, large, F₃, 21) (dual of [large, large−210, 22]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(234−21, 234, large)-Net in Base 3 — Upper bound on s

There is no (213, 234, large)-net in base 3, because

19 times m-reduction [i] would yield (213, 215, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]