MinT - Best Known (82, 82+164, s)-Nets in Base 3

Best Known (82, 82+164, s)-Nets in Base 3

(82, 82+164, 57)-Net over F₃ — Constructive and digital

Digital (82, 246, 57)-net over F₃, using

net from sequence [i] based on digital (82, 56)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 56)-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]

(82, 82+164, 84)-Net over F₃ — Digital

Digital (82, 246, 84)-net over F₃, using

t-expansion [i] based on digital (71, 246, 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]

(82, 82+164, 258)-Net over F₃ — Upper bound on s (digital)

There is no digital (82, 246, 259)-net over F₃, because

2 times m-reduction [i] would yield digital (82, 244, 259)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁴, 259, F₃, 162) (dual of [259, 15, 163]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(3²⁴³, 253, F₃, 162) (dual of [253, 10, 163]-code), but
      - constructionÂ Y1 [i] would yield
        linear OA(3²⁴², 249, F₃, 162) (dual of [249, 7, 163]-code), but
        residual code [i] would yield linear OA(3⁸⁰, 86, F₃, 54) (dual of [86, 6, 55]-code), but
        residual code [i] would yield linear OA(3²⁶, 31, F₃, 18) (dual of [31, 5, 19]-code), but
        residual code [i] would yield linear OA(3⁸, 12, F₃, 6) (dual of [12, 4, 7]-code), but
        residual code [i] would yield linear OA(3², 5, F₃, 2) (dual of [5, 3, 3]-code or 5-arc in PG(1,3)), but
        improvement by Maruta on the Griesmer bound [i]
        OA(3¹⁰, 253, 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]
    2. OA(3¹⁵, 259, S₃, 6), but
      - discarding factors would yield OA(3¹⁵, 222, S₃, 6), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 14 490089 > 3¹⁵ [i]

(82, 82+164, 347)-Net in Base 3 — Upper bound on s

There is no (82, 246, 348)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2608 091414 869786 331431 832607 934023 990173 460574 171603 472041 484089 394832 971286 198719 601321 886271 420622 838982 139910 619465 > 3²⁴⁶ [i]