MinT - Best Known (237−33, 237, s)-Nets in Base 3

Best Known (237−33, 237, s)-Nets in Base 3

(237−33, 237, 3697)-Net over F₃ — Constructive and digital

Digital (204, 237, 3697)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 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 (187, 220, 3690)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²²⁰, 3690, F₃, 33, 33) (dual of [(3690, 33), 121550, 34]-NRT-code), using
    - OOA 16-folding and stacking with additional row [i] based on linear OA(3²²⁰, 59041, F₃, 33) (dual of [59041, 58821, 34]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²²⁰, 59048, F₃, 33) (dual of [59048, 58828, 34]-code), using
        1 times truncation [i] based on linear OA(3²²¹, 59049, F₃, 34) (dual of [59049, 58828, 35]-code), using
        an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]

(237−33, 237, 29563)-Net over F₃ — Digital

Digital (204, 237, 29563)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²³⁷, 29563, F₃, 2, 33) (dual of [(29563, 2), 58889, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3²³⁷, 59126, F₃, 33) (dual of [59126, 58889, 34]-code), using
  - constructionÂ X applied to C([0,16]) âŠ‚ C([0,12]) [i] based on
    1. linear OA(3²²¹, 59050, F₃, 33) (dual of [59050, 58829, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050Â | 3²⁰âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
    2. linear OA(3¹⁶¹, 59050, F₃, 25) (dual of [59050, 58889, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050Â | 3²⁰âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    3. linear OA(3¹⁶, 76, F₃, 7) (dual of [76, 60, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁶, 82, F₃, 7) (dual of [82, 66, 8]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(237−33, 237, large)-Net in Base 3 — Upper bound on s

There is no (204, 237, large)-net in base 3, because

31 times m-reduction [i] would yield (204, 206, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]