MinT - Best Known (9−4, 9, s)-Nets in Base 8

Best Known (9−4, 9, s)-Nets in Base 8

(9−4, 9, 130)-Net over F₈ — Constructive and digital

Digital (5, 9, 130)-net over F₈, using

1 times m-reduction [i] based on digital (5, 10, 130)-net over F₈, using
- trace code for nets [i] based on digital (0, 5, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]

(9−4, 9, 163)-Net over F₈ — Digital

Digital (5, 9, 163)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁹, 163, F₈, 4, 4) (dual of [(163, 4), 643, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(8⁹, 163, F₈, 3, 4) (dual of [(163, 3), 480, 5]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁹, 163, F₈, 4) (dual of [163, 154, 5]-code), using
    - 32 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 31 times 0) [i] based on linear OA(8⁸, 130, F₈, 4) (dual of [130, 122, 5]-code), using
      - trace code [i] based on linear OA(64⁴, 65, F₆₄, 4) (dual of [65, 61, 5]-code or 65-arc in PG(3,64)), using
        extended Reedâ€“Solomon code RS_e(61,64) [i]

(9−4, 9, 2339)-Net in Base 8 — Upper bound on s

There is no (5, 9, 2340)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 134 242291 > 8⁹ [i]