MinT - Best Known (113−39, 113, s)-Nets in Base 16

Best Known (113−39, 113, s)-Nets in Base 16

(113−39, 113, 603)-Net over F₁₆ — Constructive and digital

Digital (74, 113, 603)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (16, 35, 89)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 10, 24)-net over F₁₆, using
      - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
        a function field by SÃ©mirat [i]
    2. digital (6, 25, 65)-net over F₁₆, using
      - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]
2. digital (39, 78, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 39, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(113−39, 113, 664)-Net in Base 16 — Constructive

(74, 113, 664)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (16, 35, 150)-net in base 16, using
  - base change [i] based on digital (1, 20, 150)-net over F₁₂₈, using
    - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]
2. digital (39, 78, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 39, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(113−39, 113, 4142)-Net over F₁₆ — Digital

Digital (74, 113, 4142)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹¹³, 4142, F₁₆, 39) (dual of [4142, 4029, 40]-code), using
- 35 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 5 times 0, 1, 28 times 0) [i] based on linear OA(16¹¹⁰, 4104, F₁₆, 39) (dual of [4104, 3994, 40]-code), using
  - constructionÂ X applied to C([0,19]) âŠ‚ C([0,18]) [i] based on
    1. linear OA(16¹⁰⁹, 4097, F₁₆, 39) (dual of [4097, 3988, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 16⁶âˆ’1, defining interval IÂ =Â [0,19], and minimum distance dÂ â‰¥ |{−19,−18,…,19}|+1Â =Â 40 (BCH-bound) [i]
    2. linear OA(16¹⁰³, 4097, F₁₆, 37) (dual of [4097, 3994, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 16⁶âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
    3. linear OA(16¹, 7, F₁₆, 1) (dual of [7, 6, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(16¹, s, F₁₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(113−39, 113, 6623598)-Net in Base 16 — Upper bound on s

There is no (74, 113, 6623599)-net in base 16, because

1 times m-reduction [i] would yield (74, 112, 6623599)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 726 839794 808743 238062 599601 569408 244706 435714 441515 330775 650609 579598 028447 691608 991956 512180 938330 149658 101874 741969 044782 730975 808216 > 16¹¹² [i]