MinT - Best Known (12−5, 12, s)-Nets in Base 8

Best Known (12−5, 12, s)-Nets in Base 8

(12−5, 12, 160)-Net over F₈ — Constructive and digital

Digital (7, 12, 160)-net over F₈, using

trace code for nets [i] based on digital (1, 6, 80)-net over F₆₄, using
- net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
    - a function field by SÃ©mirat [i]

(12−5, 12, 228)-Net over F₈ — Digital

Digital (7, 12, 228)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹², 228, F₈, 5) (dual of [228, 216, 6]-code), using
- generalized (u,Â u+v)-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(8¹, 57, F₈, 1) (dual of [57, 56, 2]-code) (see above)
  3. 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]
  4. linear OA(8⁷, 57, F₈, 5) (dual of [57, 50, 6]-code), using
    - a code by Danev and Olsson [i]

(12−5, 12, 496)-Net in Base 8 — Constructive

(7, 12, 496)-net in base 8, using

net defined by OOA [i] based on OOA(8¹², 496, S₈, 5, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(8¹², 993, S₈, 5), using
  - discarding parts of the base [i] based on linear OA(32⁷, 993, F₃₂, 5) (dual of [993, 986, 6]-code), using
    - a code by Danev and Olsson [i]

(12−5, 12, 18723)-Net in Base 8 — Upper bound on s

There is no (7, 12, 18724)-net in base 8, because

1 times m-reduction [i] would yield (7, 11, 18724)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 8590 131187 > 8¹¹ [i]