MinT - Best Known (242−166, 242, s)-Nets in Base 3

Best Known (242−166, 242, s)-Nets in Base 3

(242−166, 242, 51)-Net over F₃ — Constructive and digital

Digital (76, 242, 51)-net over F₃, using

net from sequence [i] based on digital (76, 50)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 50)-sequence over F₉, using
  - s-reduction based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(242−166, 242, 84)-Net over F₃ — Digital

Digital (76, 242, 84)-net over F₃, using

t-expansion [i] based on digital (71, 242, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(242−166, 242, 238)-Net over F₃ — Upper bound on s (digital)

There is no digital (76, 242, 239)-net over F₃, because

13 times m-reduction [i] would yield digital (76, 229, 239)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²²⁹, 239, F₃, 153) (dual of [239, 10, 154]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(3²²⁸, 235, F₃, 153) (dual of [235, 7, 154]-code), but
      - residual code [i] would yield linear OA(3⁷⁵, 81, F₃, 51) (dual of [81, 6, 52]-code), but
        â€œMaâ€ bound on codes from Brouwerâ€™s database [i]
    2. OA(3¹⁰, 239, S₃, 4), but
      - discarding factors would yield OA(3¹⁰, 172, S₃, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 59169 > 3¹⁰ [i]

(242−166, 242, 242)-Net in Base 3 — Upper bound on s

There is no (76, 242, 243)-net in base 3, because

4 times m-reduction [i] would yield (76, 238, 243)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3²³⁸, 243, S₃, 162), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 87 189642 485960 958202 911070 585860 771696 964072 404731 750085 525219 437990 967093 723439 943475 549906 831683 116791 055225 665627 / 163 > 3²³⁸ [i]