MinT - Best Known (31, 31+18, s)-Nets in Base 8

Best Known (31, 31+18, s)-Nets in Base 8

(31, 31+18, 256)-Net over F₈ — Constructive and digital

Digital (31, 49, 256)-net over F₈, using

3 times m-reduction [i] based on digital (31, 52, 256)-net over F₈, using
- trace code for nets [i] based on digital (5, 26, 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]

(31, 31+18, 489)-Net over F₈ — Digital

Digital (31, 49, 489)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁴⁹, 489, F₈, 18) (dual of [489, 440, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁴⁹, 511, F₈, 18) (dual of [511, 462, 19]-code), using
  - the primitive BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â {−5,−4,…,12}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(31, 31+18, 514)-Net in Base 8 — Constructive

(31, 49, 514)-net in base 8, using

8¹ times duplication [i] based on (30, 48, 514)-net in base 8, using
- base change [i] based on digital (18, 36, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 18, 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]

(31, 31+18, 48913)-Net in Base 8 — Upper bound on s

There is no (31, 49, 48914)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 178 418661 303753 000668 159392 776837 321281 304906 > 8⁴⁹ [i]