MinT - Best Known (41−7, 41, s)-Nets in Base 8

Best Known (41−7, 41, s)-Nets in Base 8

(41−7, 41, 174773)-Net over F₈ — Constructive and digital

Digital (34, 41, 174773)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 3, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. digital (31, 38, 174764)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8³⁸, 174764, F₈, 7, 7) (dual of [(174764, 7), 1223310, 8]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OA(8³⁸, 524293, F₈, 7) (dual of [524293, 524255, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(8³⁸, 524294, F₈, 7) (dual of [524294, 524256, 8]-code), using
        trace code [i] based on linear OA(64¹⁹, 262147, F₆₄, 7) (dual of [262147, 262128, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(64¹⁹, 262144, F₆₄, 7) (dual of [262144, 262125, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(64¹⁶, 262144, F₆₄, 6) (dual of [262144, 262128, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 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]

(41−7, 41, 634221)-Net over F₈ — Digital

Digital (34, 41, 634221)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(41−7, 41, large)-Net in Base 8 — Upper bound on s

There is no (34, 41, large)-net in base 8, because

5 times m-reduction [i] would yield (34, 36, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (41−7, 41, s)-Nets in Base 8

(41−7, 41, 174773)-Net over F8 — Constructive and digital

(41−7, 41, 634221)-Net over F8 — Digital

(41−7, 41, large)-Net in Base 8 — Upper bound on s

(41−7, 41, 174773)-Net over F₈ — Constructive and digital

(41−7, 41, 634221)-Net over F₈ — Digital