MinT - Best Known (113−20, 113, s)-Nets in Base 8

Best Known (113−20, 113, s)-Nets in Base 8

(113−20, 113, 26224)-Net over F₈ — Constructive and digital

Digital (93, 113, 26224)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 10, 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 (83, 103, 26215)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁰³, 26215, F₈, 20, 20) (dual of [(26215, 20), 524197, 21]-NRT-code), using
    - OA 10-folding and stacking [i] based on linear OA(8¹⁰³, 262150, F₈, 20) (dual of [262150, 262047, 21]-code), using
      - constructionÂ X applied to C_e(19) âŠ‚ C_e(18) [i] based on
        linear OA(8¹⁰³, 262144, F₈, 20) (dual of [262144, 262041, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(8⁹⁷, 262144, F₈, 19) (dual of [262144, 262047, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
        linear OA(8⁰, 6, F₈, 0) (dual of [6, 6, 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]

(113−20, 113, 266159)-Net over F₈ — Digital

Digital (93, 113, 266159)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(113−20, 113, large)-Net in Base 8 — Upper bound on s

There is no (93, 113, large)-net in base 8, because

18 times m-reduction [i] would yield (93, 95, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (113−20, 113, s)-Nets in Base 8

(113−20, 113, 26224)-Net over F8 — Constructive and digital

(113−20, 113, 266159)-Net over F8 — Digital

(113−20, 113, large)-Net in Base 8 — Upper bound on s

(113−20, 113, 26224)-Net over F₈ — Constructive and digital

(113−20, 113, 266159)-Net over F₈ — Digital