MinT - Best Known (19, 30, s)-Nets in Base 8

Best Known (19, 30, s)-Nets in Base 8

(19, 30, 208)-Net over F₈ — Constructive and digital

Digital (19, 30, 208)-net over F₈, using

2 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]

(19, 30, 477)-Net over F₈ — Digital

Digital (19, 30, 477)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8³⁰, 477, F₈, 11) (dual of [477, 447, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(8³⁰, 511, F₈, 11) (dual of [511, 481, 12]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(19, 30, 514)-Net in Base 8 — Constructive

(19, 30, 514)-net in base 8, using

trace code for nets [i] based on (4, 15, 257)-net in base 64, using
- 1 times m-reduction [i] based on (4, 16, 257)-net in base 64, using
  - base change [i] based on digital (0, 12, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(19, 30, 64363)-Net in Base 8 — Upper bound on s

There is no (19, 30, 64364)-net in base 8, because

1 times m-reduction [i] would yield (19, 29, 64364)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 154 744654 049684 535559 760682 > 8²⁹ [i]