MinT - Best Known (31−12, 31, s)-Nets in Base 8

Best Known (31−12, 31, s)-Nets in Base 8

(31−12, 31, 208)-Net over F₈ — Constructive and digital

Digital (19, 31, 208)-net over F₈, using

1 times m-reduction [i] based on digital (19, 32, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 16, 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]

(31−12, 31, 258)-Net in Base 8 — Constructive

(19, 31, 258)-net in base 8, using

1 times m-reduction [i] based on (19, 32, 258)-net in base 8, using
- trace code for nets [i] based on (3, 16, 129)-net in base 64, using
  - 5 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]

(31−12, 31, 326)-Net over F₈ — Digital

Digital (19, 31, 326)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8³¹, 326, F₈, 12) (dual of [326, 295, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(8³¹, 511, F₈, 12) (dual of [511, 480, 13]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(31−12, 31, 19815)-Net in Base 8 — Upper bound on s

There is no (19, 31, 19816)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 9903 600197 335615 601989 920847 > 8³¹ [i]