MinT - Best Known (65−9, 65, s)-Nets in Base 8

Best Known (65−9, 65, s)-Nets in Base 8

(65−9, 65, 2097280)-Net over F₈ — Constructive and digital

Digital (56, 65, 2097280)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 8, 130)-net over F₈, using
  - trace code for nets [i] based on digital (0, 4, 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]
2. digital (48, 57, 2097150)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁵⁷, 2097150, F₈, 9, 9) (dual of [(2097150, 9), 18874293, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(8⁵⁷, 8388601, F₈, 9) (dual of [8388601, 8388544, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁵⁷, large, F₈, 9) (dual of [large, large−57, 10]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(65−9, 65, large)-Net over F₈ — Digital

Digital (56, 65, large)-net over F₈, using

t-expansion [i] based on digital (55, 65, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶⁵, large, F₈, 10) (dual of [large, large−65, 11]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(65−9, 65, large)-Net in Base 8 — Upper bound on s

There is no (56, 65, large)-net in base 8, because

7 times m-reduction [i] would yield (56, 58, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]