MinT - Best Known (75, 75+41, s)-Nets in Base 16

Best Known (75, 75+41, s)-Nets in Base 16

(75, 75+41, 587)-Net over F₁₆ — Constructive and digital

Digital (75, 116, 587)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 26, 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 (49, 90, 522)-net over F₁₆, using
  - trace code for nets [i] based on digital (4, 45, 261)-net over F₂₅₆, using
    - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(75, 75+41, 612)-Net in Base 16 — Constructive

(75, 116, 612)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (14, 34, 98)-net in base 16, using
  - 1 times m-reduction [i] based on (14, 35, 98)-net in base 16, using
    - base change [i] based on digital (7, 28, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]
2. digital (41, 82, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 41, 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]

(75, 75+41, 3626)-Net over F₁₆ — Digital

Digital (75, 116, 3626)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹¹⁶, 3626, F₁₆, 41) (dual of [3626, 3510, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(16¹¹⁶, 4104, F₁₆, 41) (dual of [4104, 3988, 42]-code), using
  - constructionÂ X applied to C([0,20]) âŠ‚ C([0,19]) [i] based on
    1. linear OA(16¹¹⁵, 4097, F₁₆, 41) (dual of [4097, 3982, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 16⁶âˆ’1, defining interval IÂ =Â [0,20], and minimum distance dÂ â‰¥ |{−20,−19,…,20}|+1Â =Â 42 (BCH-bound) [i]
    2. 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]
    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]

(75, 75+41, 4644124)-Net in Base 16 — Upper bound on s

There is no (75, 116, 4644125)-net in base 16, because

1 times m-reduction [i] would yield (75, 115, 4644125)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 977137 116940 136118 260416 505329 922891 065866 528492 941380 332923 640914 369513 327118 245751 343590 791394 704253 215793 902361 302507 369462 397898 675001 > 16¹¹⁵ [i]