MinT - Best Known (73, 113, s)-Nets in Base 16

Best Known (73, 113, s)-Nets in Base 16

(73, 113, 585)-Net over F₁₆ — Constructive and digital

Digital (73, 113, 585)-net over F₁₆, using

1 times m-reduction [i] based on digital (73, 114, 585)-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 (47, 88, 520)-net over F₁₆, using
    - trace code for nets [i] based on digital (3, 44, 260)-net over F₂₅₆, using
      - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(73, 113, 594)-Net in Base 16 — Constructive

(73, 113, 594)-net in base 16, using

1 times m-reduction [i] based on (73, 114, 594)-net in base 16, using
- (u,Â u+v)-construction [i] based on
  1. (12, 32, 80)-net in base 16, using
    - 1 times m-reduction [i] based on (12, 33, 80)-net in base 16, using
      - base change [i] based on digital (1, 22, 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 (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]

(73, 113, 3526)-Net over F₁₆ — Digital

Digital (73, 113, 3526)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹¹³, 3526, F₁₆, 40) (dual of [3526, 3413, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(16¹¹³, 4103, F₁₆, 40) (dual of [4103, 3990, 41]-code), using
  - constructionÂ X applied to C_e(39) âŠ‚ C_e(37) [i] based on
    1. linear OA(16¹¹², 4096, F₁₆, 40) (dual of [4096, 3984, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
    2. linear OA(16¹⁰⁶, 4096, F₁₆, 38) (dual of [4096, 3990, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [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]

(73, 113, 3519585)-Net in Base 16 — Upper bound on s

There is no (73, 113, 3519586)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 11629 443414 416692 068302 675236 700500 626972 056669 813241 071848 389458 600703 360767 803645 645472 317816 929873 693775 030533 422970 914277 083142 329176 > 16¹¹³ [i]