MinT - Best Known (59−21, 59, s)-Nets in Base 16

Best Known (59−21, 59, s)-Nets in Base 16

(59−21, 59, 579)-Net over F₁₆ — Constructive and digital

Digital (38, 59, 579)-net over F₁₆, using

16¹ times duplication [i] based on digital (37, 58, 579)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 16, 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 (21, 42, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 21, 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]

(59−21, 59, 594)-Net in Base 16 — Constructive

(38, 59, 594)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (7, 17, 80)-net in base 16, using
  - 1 times m-reduction [i] based on (7, 18, 80)-net in base 16, using
    - base change [i] based on digital (1, 12, 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 (21, 42, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 21, 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]

(59−21, 59, 2496)-Net over F₁₆ — Digital

Digital (38, 59, 2496)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁵⁹, 2496, F₁₆, 21) (dual of [2496, 2437, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁵⁹, 4103, F₁₆, 21) (dual of [4103, 4044, 22]-code), using
  - constructionÂ X applied to C_e(20) âŠ‚ C_e(18) [i] based on
    1. linear OA(16⁵⁸, 4096, F₁₆, 21) (dual of [4096, 4038, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
    2. linear OA(16⁵², 4096, F₁₆, 19) (dual of [4096, 4044, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [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]

(59−21, 59, 2909244)-Net in Base 16 — Upper bound on s

There is no (38, 59, 2909245)-net in base 16, because

1 times m-reduction [i] would yield (38, 58, 2909245)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 6901 766090 086726 542641 998669 665191 287620 569306 698811 046114 058924 966126 > 16⁵⁸ [i]