MinT - Best Known (67, 89, s)-Nets in Base 8

Best Known (67, 89, s)-Nets in Base 8

(67, 89, 745)-Net over F₈ — Constructive and digital

Digital (67, 89, 745)-net over F₈, using

1 times m-reduction [i] based on 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
        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]
        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]
        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]

(67, 89, 1028)-Net in Base 8 — Constructive

(67, 89, 1028)-net in base 8, using

8¹ times duplication [i] based on (66, 88, 1028)-net in base 8, using
- base change [i] based on digital (44, 66, 1028)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (11, 22, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 11, 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 (22, 44, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 22, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(67, 89, 8343)-Net over F₈ — Digital

Digital (67, 89, 8343)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 89, large)-Net in Base 8 — Upper bound on s

There is no (67, 89, large)-net in base 8, because

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