MinT - Best Known (101−25, 101, s)-Nets in Base 8

Best Known (101−25, 101, s)-Nets in Base 8

(101−25, 101, 683)-Net over F₈ — Constructive and digital

Digital (76, 101, 683)-net over F₈, using

8² times duplication [i] based on digital (74, 99, 683)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8⁹⁹, 683, F₈, 25, 25) (dual of [(683, 25), 16976, 26]-NRT-code), using
  - OOA 12-folding and stacking with additional row [i] based on linear OA(8⁹⁹, 8197, F₈, 25) (dual of [8197, 8098, 26]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(8⁹⁸, 8196, F₈, 25) (dual of [8196, 8098, 26]-code), using
      - trace code [i] based on linear OA(64⁴⁹, 4098, F₆₄, 25) (dual of [4098, 4049, 26]-code), using
        constructionÂ X applied to C_e(24) âŠ‚ C_e(23) [i] based on
        linear OA(64⁴⁹, 4096, F₆₄, 25) (dual of [4096, 4047, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
        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]
        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]

(101−25, 101, 1028)-Net in Base 8 — Constructive

(76, 101, 1028)-net in base 8, using

8¹ times duplication [i] based on (75, 100, 1028)-net in base 8, using
- base change [i] based on digital (50, 75, 1028)-net over F₁₆, using
  - 16¹ times duplication [i] based on digital (49, 74, 1028)-net over F₁₆, using
    - (u,Â u+v)-construction [i] based on
      1. digital (12, 24, 514)-net over F₁₆, using
        trace code for nets [i] based on digital (0, 12, 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 (25, 50, 514)-net over F₁₆, using
        trace code for nets [i] based on digital (0, 25, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(101−25, 101, 8858)-Net over F₈ — Digital

Digital (76, 101, 8858)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(101−25, 101, large)-Net in Base 8 — Upper bound on s

There is no (76, 101, large)-net in base 8, because

23 times m-reduction [i] would yield (76, 78, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (101−25, 101, s)-Nets in Base 8

(101−25, 101, 683)-Net over F8 — Constructive and digital

(101−25, 101, 1028)-Net in Base 8 — Constructive

(101−25, 101, 8858)-Net over F8 — Digital

(101−25, 101, large)-Net in Base 8 — Upper bound on s

(101−25, 101, 683)-Net over F₈ — Constructive and digital

(101−25, 101, 8858)-Net over F₈ — Digital