MinT - Best Known (90−23, 90, s)-Nets in Base 8

Best Known (90−23, 90, s)-Nets in Base 8

(90−23, 90, 745)-Net over F₈ — Constructive and digital

Digital (67, 90, 745)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁹⁰, 745, F₈, 23, 23) (dual of [(745, 23), 17045, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8⁹⁰, 8196, F₈, 23) (dual of [8196, 8106, 24]-code), using
  - trace code [i] based on linear OA(64⁴⁵, 4098, F₆₄, 23) (dual of [4098, 4053, 24]-code), using
    - constructionÂ X applied to C_e(22) âŠ‚ C_e(21) [i] based on
      1. 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]
      2. linear OA(64⁴³, 4096, F₆₄, 22) (dual of [4096, 4053, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [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]

(90−23, 90, 814)-Net in Base 8 — Constructive

(67, 90, 814)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (17, 28, 300)-net in base 8, using
  - trace code for nets [i] based on (3, 14, 150)-net in base 64, using
    - base change [i] based on digital (1, 12, 150)-net over F₁₂₈, using
      - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]
2. (39, 62, 514)-net in base 8, using
  - trace code for nets [i] based on (8, 31, 257)-net in base 64, using
    - 1 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
      - base change [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]

(90−23, 90, 8196)-Net over F₈ — Digital

Digital (67, 90, 8196)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁹⁰, 8196, F₈, 23) (dual of [8196, 8106, 24]-code), using
- trace code [i] based on linear OA(64⁴⁵, 4098, F₆₄, 23) (dual of [4098, 4053, 24]-code), using
  - constructionÂ X applied to C_e(22) âŠ‚ C_e(21) [i] based on
    1. 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]
    2. linear OA(64⁴³, 4096, F₆₄, 22) (dual of [4096, 4053, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [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]

(90−23, 90, large)-Net in Base 8 — Upper bound on s

There is no (67, 90, large)-net in base 8, because

21 times m-reduction [i] would yield (67, 69, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]