MinT - Best Known (59−12, 59, s)-Nets in Base 8

Best Known (59−12, 59, s)-Nets in Base 8

(59−12, 59, 5479)-Net over F₈ — Constructive and digital

Digital (47, 59, 5479)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 8, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. digital (39, 51, 5462)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁵¹, 5462, F₈, 12, 12) (dual of [(5462, 12), 65493, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(8⁵¹, 32772, F₈, 12) (dual of [32772, 32721, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁵¹, 32773, F₈, 12) (dual of [32773, 32722, 13]-code), using
        constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
        linear OA(8⁵¹, 32768, F₈, 12) (dual of [32768, 32717, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(8⁴⁶, 32768, F₈, 11) (dual of [32768, 32722, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(8⁰, 5, F₈, 0) (dual of [5, 5, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(59−12, 59, 48956)-Net over F₈ — Digital

Digital (47, 59, 48956)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(59−12, 59, large)-Net in Base 8 — Upper bound on s

There is no (47, 59, large)-net in base 8, because

10 times m-reduction [i] would yield (47, 49, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (59−12, 59, s)-Nets in Base 8

(59−12, 59, 5479)-Net over F8 — Constructive and digital

(59−12, 59, 48956)-Net over F8 — Digital

(59−12, 59, large)-Net in Base 8 — Upper bound on s

(59−12, 59, 5479)-Net over F₈ — Constructive and digital

(59−12, 59, 48956)-Net over F₈ — Digital