MinT - Best Known (6, 6+5, s)-Nets in Base 8

Best Known (6, 6+5, s)-Nets in Base 8

(6, 6+5, 130)-Net over F₈ — Constructive and digital

Digital (6, 11, 130)-net over F₈, using

1 times m-reduction [i] based on digital (6, 12, 130)-net over F₈, using
- trace code for nets [i] based on digital (0, 6, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]

(6, 6+5, 171)-Net over F₈ — Digital

Digital (6, 11, 171)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹, 171, F₈, 5) (dual of [171, 160, 6]-code), using
- (u,Â uâˆ’v,Â u+v+w)-construction [i] based on
  1. linear OA(8¹, 57, F₈, 1) (dual of [57, 56, 2]-code), using
    - discarding factors / shortening the dual code based on linear OA(8¹, s, F₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
      - parity-check code [i]
  2. linear OA(8³, 57, F₈, 2) (dual of [57, 54, 3]-code), using
    - discarding factors / shortening the dual code based on linear OA(8³, 63, F₈, 2) (dual of [63, 60, 3]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 8²âˆ’1, defining interval IÂ =Â [0,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
  3. linear OA(8⁷, 57, F₈, 5) (dual of [57, 50, 6]-code), using
    - a code by Danev and Olsson [i]

(6, 6+5, 6619)-Net in Base 8 — Upper bound on s

There is no (6, 11, 6620)-net in base 8, because

1 times m-reduction [i] would yield (6, 10, 6620)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1073 952671 > 8¹⁰ [i]