MinT - Best Known (42, 67, s)-Nets in Base 8

Best Known (42, 67, s)-Nets in Base 8

(42, 67, 354)-Net over F₈ — Constructive and digital

Digital (42, 67, 354)-net over F₈, using

3 times m-reduction [i] based on digital (42, 70, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 35, 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]

(42, 67, 384)-Net in Base 8 — Constructive

(42, 67, 384)-net in base 8, using

1 times m-reduction [i] based on (42, 68, 384)-net in base 8, using
- trace code for nets [i] based on (8, 34, 192)-net in base 64, using
  - 1 times m-reduction [i] based on (8, 35, 192)-net in base 64, using
    - base change [i] based on digital (3, 30, 192)-net over F₁₂₈, using
      - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(42, 67, 513)-Net over F₈ — Digital

Digital (42, 67, 513)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶⁷, 513, F₈, 25) (dual of [513, 446, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 513Â | 8⁶âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(42, 67, 70018)-Net in Base 8 — Upper bound on s

There is no (42, 67, 70019)-net in base 8, because

1 times m-reduction [i] would yield (42, 66, 70019)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 401760 540233 693384 926938 142831 820000 412963 224826 931230 412064 > 8⁶⁶ [i]