MinT - Best Known (61−23, 61, s)-Nets in Base 8

Best Known (61−23, 61, s)-Nets in Base 8

(61−23, 61, 354)-Net over F₈ — Constructive and digital

Digital (38, 61, 354)-net over F₈, using

1 times m-reduction [i] based on digital (38, 62, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 31, 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]

(61−23, 61, 460)-Net over F₈ — Digital

Digital (38, 61, 460)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶¹, 460, F₈, 23) (dual of [460, 399, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁶¹, 511, F₈, 23) (dual of [511, 450, 24]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(61−23, 61, 59130)-Net in Base 8 — Upper bound on s

There is no (38, 61, 59131)-net in base 8, because

1 times m-reduction [i] would yield (38, 60, 59131)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 532572 557642 541131 623474 088370 912552 729063 970381 491928 > 8⁶⁰ [i]