MinT - Best Known (85, 118, s)-Nets in Base 8

Best Known (85, 118, s)-Nets in Base 8

(85, 118, 562)-Net over F₈ — Constructive and digital

Digital (85, 118, 562)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (22, 38, 208)-net over F₈, using
  - trace code for nets [i] based on digital (3, 19, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]
2. digital (47, 80, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 40, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(85, 118, 612)-Net in Base 8 — Constructive

(85, 118, 612)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (22, 38, 258)-net in base 8, using
  - trace code for nets [i] based on (3, 19, 129)-net in base 64, using
    - 2 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
      - base change [i] based on digital (0, 18, 129)-net over F₁₂₈, using
        net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
2. digital (47, 80, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 40, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(85, 118, 4180)-Net over F₈ — Digital

Digital (85, 118, 4180)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹⁸, 4180, F₈, 33) (dual of [4180, 4062, 34]-code), using
- 78 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 6 times 0, 1, 18 times 0, 1, 49 times 0) [i] based on linear OA(8¹¹³, 4097, F₈, 33) (dual of [4097, 3984, 34]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]

(85, 118, 3901980)-Net in Base 8 — Upper bound on s

There is no (85, 118, 3901981)-net in base 8, because

1 times m-reduction [i] would yield (85, 117, 3901981)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 4587 001461 231488 901278 694289 453065 211495 453057 306969 999491 112151 114212 910457 606345 241438 531162 996172 803483 > 8¹¹⁷ [i]