MinT - Best Known (166−18, 166, s)-Nets in Base 8

Best Known (166−18, 166, s)-Nets in Base 8

(166−18, 166, 1864424)-Net over F₈ — Constructive and digital

Digital (148, 166, 1864424)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (19, 28, 290)-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 (11, 20, 160)-net over F₈, using
      - trace code for nets [i] based on digital (1, 10, 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]
2. digital (120, 138, 1864134)-net over F₈, using
  - trace code for nets [i] based on digital (51, 69, 932067)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁶⁹, 932067, F₆₄, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
      - OA 9-folding and stacking [i] based on linear OA(64⁶⁹, large, F₆₄, 18) (dual of [large, large−69, 19]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(166−18, 166, 1864657)-Net in Base 8 — Constructive

(148, 166, 1864657)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (19, 28, 523)-net in base 8, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 4, 9)-net over F₈, using
      - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
        the rational function field F₈(x) [i]
        Niederreiter sequence [i]
    2. (15, 24, 514)-net in base 8, using
      - base change [i] based on digital (9, 18, 514)-net over F₁₆, using
        trace code for nets [i] based on digital (0, 9, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
2. digital (120, 138, 1864134)-net over F₈, using
  - trace code for nets [i] based on digital (51, 69, 932067)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁶⁹, 932067, F₆₄, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
      - OA 9-folding and stacking [i] based on linear OA(64⁶⁹, large, F₆₄, 18) (dual of [large, large−69, 19]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(166−18, 166, large)-Net over F₈ — Digital

Digital (148, 166, large)-net over F₈, using

t-expansion [i] based on digital (144, 166, large)-net over F₈, using
- 2 times m-reduction [i] based on digital (144, 168, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶⁸, large, F₈, 24) (dual of [large, large−168, 25]-code), using
    - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(166−18, 166, large)-Net in Base 8 — Upper bound on s

There is no (148, 166, large)-net in base 8, because

16 times m-reduction [i] would yield (148, 150, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (166−18, 166, s)-Nets in Base 8

(166−18, 166, 1864424)-Net over F8 — Constructive and digital

(166−18, 166, 1864657)-Net in Base 8 — Constructive

(166−18, 166, large)-Net over F8 — Digital

(166−18, 166, large)-Net in Base 8 — Upper bound on s

(166−18, 166, 1864424)-Net over F₈ — Constructive and digital

(166−18, 166, large)-Net over F₈ — Digital