MinT - Best Known (67, 67+37, s)-Nets in Base 16

Best Known (67, 67+37, s)-Nets in Base 16

(67, 67+37, 585)-Net over F₁₆ — Constructive and digital

Digital (67, 104, 585)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 24, 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 (43, 80, 520)-net over F₁₆, using
  - trace code for nets [i] based on digital (3, 40, 260)-net over F₂₅₆, using
    - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(67, 67+37, 594)-Net in Base 16 — Constructive

(67, 104, 594)-net in base 16, using

16¹ times duplication [i] based on (66, 103, 594)-net in base 16, using
- (u,Â u+v)-construction [i] based on
  1. (11, 29, 80)-net in base 16, using
    - 1 times m-reduction [i] based on (11, 30, 80)-net in base 16, using
      - base change [i] based on digital (1, 20, 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 (37, 74, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 37, 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]

(67, 67+37, 3223)-Net over F₁₆ — Digital

Digital (67, 104, 3223)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹⁰⁴, 3223, F₁₆, 37) (dual of [3223, 3119, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(16¹⁰⁴, 4104, F₁₆, 37) (dual of [4104, 4000, 38]-code), using
  - constructionÂ X applied to C([0,18]) âŠ‚ C([0,17]) [i] based on
    1. 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]
    2. linear OA(16⁹⁷, 4097, F₁₆, 35) (dual of [4097, 4000, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 16⁶âˆ’1, defining interval IÂ =Â [0,17], and minimum distance dÂ â‰¥ |{−17,−16,…,17}|+1Â =Â 36 (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]

(67, 67+37, 3910928)-Net in Base 16 — Upper bound on s

There is no (67, 104, 3910929)-net in base 16, because

1 times m-reduction [i] would yield (67, 103, 3910929)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 10576 934199 609779 750012 143563 571932 678550 693513 370424 349786 645851 476555 325123 338546 354892 961959 769380 744221 945540 642616 739456 > 16¹⁰³ [i]