MinT - Best Known (94−24, 94, s)-Nets in Base 8

Best Known (94−24, 94, s)-Nets in Base 8

(94−24, 94, 683)-Net over F₈ — Constructive and digital

Digital (70, 94, 683)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁹⁴, 683, F₈, 24, 24) (dual of [(683, 24), 16298, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(8⁹⁴, 8196, F₈, 24) (dual of [8196, 8102, 25]-code), using
  - trace code [i] based on linear OA(64⁴⁷, 4098, F₆₄, 24) (dual of [4098, 4051, 25]-code), using
    - constructionÂ X applied to C_e(23) âŠ‚ C_e(22) [i] based on
      1. linear OA(64⁴⁷, 4096, F₆₄, 24) (dual of [4096, 4049, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]
      2. linear OA(64⁴⁵, 4096, F₆₄, 23) (dual of [4096, 4051, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      3. linear OA(64⁰, 2, F₆₄, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(94−24, 94, 772)-Net in Base 8 — Constructive

(70, 94, 772)-net in base 8, using

8² times duplication [i] based on (68, 92, 772)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. (16, 28, 258)-net in base 8, using
    - trace code for nets [i] based on (2, 14, 129)-net in base 64, using
      - base change [i] based on digital (0, 12, 129)-net over F₁₂₈, using
        net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
  2. (40, 64, 514)-net in base 8, using
    - base change [i] based on digital (24, 48, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 24, 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]

(94−24, 94, 8196)-Net over F₈ — Digital

Digital (70, 94, 8196)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁹⁴, 8196, F₈, 24) (dual of [8196, 8102, 25]-code), using
- trace code [i] based on linear OA(64⁴⁷, 4098, F₆₄, 24) (dual of [4098, 4051, 25]-code), using
  - constructionÂ X applied to C_e(23) âŠ‚ C_e(22) [i] based on
    1. linear OA(64⁴⁷, 4096, F₆₄, 24) (dual of [4096, 4049, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]
    2. linear OA(64⁴⁵, 4096, F₆₄, 23) (dual of [4096, 4051, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    3. linear OA(64⁰, 2, F₆₄, 0) (dual of [2, 2, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(94−24, 94, large)-Net in Base 8 — Upper bound on s

There is no (70, 94, large)-net in base 8, because

22 times m-reduction [i] would yield (70, 72, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]