MinT - Best Known (116−33, 116, s)-Nets in Base 8

Best Known (116−33, 116, s)-Nets in Base 8

(116−33, 116, 514)-Net over F₈ — Constructive and digital

Digital (83, 116, 514)-net over F₈, using

8² times duplication [i] based on digital (81, 114, 514)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (18, 34, 160)-net over F₈, using
    - trace code for nets [i] based on digital (1, 17, 80)-net over F₆₄, using
      - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
  2. digital (47, 80, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 40, 177)-net over F₆₄, using
      - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(116−33, 116, 593)-Net in Base 8 — Constructive

(83, 116, 593)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (2, 18, 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. (65, 98, 576)-net in base 8, using
  - trace code for nets [i] based on (16, 49, 288)-net in base 64, using
    - base change [i] based on digital (9, 42, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(116−33, 116, 3954)-Net over F₈ — Digital

Digital (83, 116, 3954)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹⁶, 3954, F₈, 33) (dual of [3954, 3838, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(8¹¹⁶, 4107, F₈, 33) (dual of [4107, 3991, 34]-code), using
  - constructionÂ X applied to C([0,16]) âŠ‚ C([0,14]) [i] based on
    1. linear OA(8¹¹³, 4097, F₈, 33) (dual of [4097, 3984, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
    2. linear OA(8¹⁰⁵, 4097, F₈, 29) (dual of [4097, 3992, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,14], and minimum distance dÂ â‰¥ |{−14,−13,…,14}|+1Â =Â 30 (BCH-bound) [i]
    3. linear OA(8³, 10, F₈, 3) (dual of [10, 7, 4]-code or 10-arc in PG(2,8) or 10-cap in PG(2,8)), using
      - hyperoval in PG(2,Â 8) [i]

(116−33, 116, 3008836)-Net in Base 8 — Upper bound on s

There is no (83, 116, 3008837)-net in base 8, because

1 times m-reduction [i] would yield (83, 115, 3008837)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 71 672115 274088 034177 386792 578184 306466 902470 428855 339637 219291 227811 811910 931254 525258 643860 830035 330140 > 8¹¹⁵ [i]