MinT - Best Known (164−18, 164, s)-Nets in Base 8

Best Known (164−18, 164, s)-Nets in Base 8

(164−18, 164, 1864394)-Net over F₈ — Constructive and digital

Digital (146, 164, 1864394)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (17, 26, 260)-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 (9, 18, 130)-net over F₈, using
      - trace code for nets [i] based on digital (0, 9, 65)-net over F₆₄, using
        net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
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]

(164−18, 164, 1864648)-Net in Base 8 — Constructive

(146, 164, 1864648)-net in base 8, using

base change [i] based on digital (105, 123, 1864648)-net over F₁₆, using
- 16¹ times duplication [i] based on digital (104, 122, 1864648)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. 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 (86, 104, 1864134)-net over F₁₆, using
      - trace code for nets [i] based on digital (34, 52, 932067)-net over F₂₅₆, using
        net defined by OOA [i] based on linear OOA(256⁵², 932067, F₂₅₆, 18, 18) (dual of [(932067, 18), 16777154, 19]-NRT-code), using
        OA 9-folding and stacking [i] based on linear OA(256⁵², large, F₂₅₆, 18) (dual of [large, large−52, 19]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(164−18, 164, large)-Net over F₈ — Digital

Digital (146, 164, large)-net over F₈, using

t-expansion [i] based on digital (144, 164, large)-net over F₈, using
- 4 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]

(164−18, 164, large)-Net in Base 8 — Upper bound on s

There is no (146, 164, large)-net in base 8, because

16 times m-reduction [i] would yield (146, 148, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]