MinT - Best Known (65−27, 65, s)-Nets in Base 9

Best Known (65−27, 65, s)-Nets in Base 9

(65−27, 65, 320)-Net over F₉ — Constructive and digital

Digital (38, 65, 320)-net over F₉, using

1 times m-reduction [i] based on digital (38, 66, 320)-net over F₉, using
- trace code for nets [i] based on digital (5, 33, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(65−27, 65, 340)-Net over F₉ — Digital

Digital (38, 65, 340)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁶⁵, 340, F₉, 27) (dual of [340, 275, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(9⁶⁵, 342, F₉, 27) (dual of [342, 277, 28]-code), using
  - 7 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 6 times 0) [i] based on linear OA(9⁶⁴, 334, F₉, 27) (dual of [334, 270, 28]-code), using
    - trace code [i] based on linear OA(81³², 167, F₈₁, 27) (dual of [167, 135, 28]-code), using
      - extended algebraic-geometric code AG_e(F,139P) [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 167, using
        a curve from the manYPoints database [i]

(65−27, 65, 35320)-Net in Base 9 — Upper bound on s

There is no (38, 65, 35321)-net in base 9, because

1 times m-reduction [i] would yield (38, 64, 35321)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 11 790629 345402 050290 888814 482819 620077 709702 356059 629055 541929 > 9⁶⁴ [i]