MinT - Best Known (31, 31+16, s)-Nets in Base 8

Best Known (31, 31+16, s)-Nets in Base 8

Digital (31, 47, 354)-net over F₈, using

1 times m-reduction [i] based on digital (31, 48, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 24, 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]

(31, 47, 516)-net in base 8, using

1 times m-reduction [i] based on (31, 48, 516)-net in base 8, using
- base change [i] based on digital (19, 36, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 18, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

Digital (31, 47, 635)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁴⁷, 635, F₈, 16) (dual of [635, 588, 17]-code), using
- 118 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 4 times 0, 1, 14 times 0, 1, 34 times 0, 1, 62 times 0) [i] based on linear OA(8⁴², 512, F₈, 16) (dual of [512, 470, 17]-code), using
  - 1 times truncation [i] based on linear OA(8⁴³, 513, F₈, 17) (dual of [513, 470, 18]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 513Â | 8⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

There is no (31, 47, 108700)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2 787716 096724 949287 315201 510073 002583 799131 > 8⁴⁷ [i]