MinT - Best Known (40−15, 40, s)-Nets in Base 8

Best Known (40−15, 40, s)-Nets in Base 8

(40−15, 40, 256)-Net over F₈ — Constructive and digital

Digital (25, 40, 256)-net over F₈, using

trace code for nets [i] based on digital (5, 20, 128)-net over F₆₄, using
- net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
    - a function field by SÃ©mirat [i]

(40−15, 40, 408)-Net over F₈ — Digital

Digital (25, 40, 408)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁴⁰, 408, F₈, 15) (dual of [408, 368, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁴⁰, 511, F₈, 15) (dual of [511, 471, 16]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(40−15, 40, 514)-Net in Base 8 — Constructive

(25, 40, 514)-net in base 8, using

base change [i] based on digital (15, 30, 514)-net over F₁₆, using
- trace code for nets [i] based on digital (0, 15, 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]

(40−15, 40, 51914)-Net in Base 8 — Upper bound on s

There is no (25, 40, 51915)-net in base 8, because

1 times m-reduction [i] would yield (25, 39, 51915)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 166165 160244 849161 437599 503316 539512 > 8³⁹ [i]