MinT - Best Known (109−13, 109, s)-Nets in Base 8

Best Known (109−13, 109, s)-Nets in Base 8

(109−13, 109, 2796265)-Net over F₈ — Constructive and digital

Digital (96, 109, 2796265)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 11, 65)-net over F₈, using
  - base reduction for projective spaces (embedding PG(5,64) in PG(10,8)) for nets [i] based on digital (0, 6, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
      - Niederreiter sequence [i]
2. digital (85, 98, 2796200)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁹⁸, 2796200, F₈, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(8⁹⁸, 8388601, F₈, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁹⁸, 8388602, F₈, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
        trace code [i] based on linear OOA(64⁴⁹, 4194301, F₆₄, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁴⁹, 8388602, F₆₄, 13) (dual of [8388602, 8388553, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴⁹, large, F₆₄, 13) (dual of [large, large−49, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(109−13, 109, large)-Net over F₈ — Digital

Digital (96, 109, large)-net over F₈, using

3 times m-reduction [i] based on digital (96, 112, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹², large, F₈, 16) (dual of [large, large−112, 17]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(109−13, 109, large)-Net in Base 8 — Upper bound on s

There is no (96, 109, large)-net in base 8, because

11 times m-reduction [i] would yield (96, 98, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (109−13, 109, s)-Nets in Base 8

(109−13, 109, 2796265)-Net over F8 — Constructive and digital

(109−13, 109, large)-Net over F8 — Digital

(109−13, 109, large)-Net in Base 8 — Upper bound on s

(109−13, 109, 2796265)-Net over F₈ — Constructive and digital

(109−13, 109, large)-Net over F₈ — Digital