MinT - Best Known (91−32, 91, s)-Nets in Base 16

Best Known (91−32, 91, s)-Nets in Base 16

(91−32, 91, 583)-Net over F₁₆ — Constructive and digital

Digital (59, 91, 583)-net over F₁₆, using

1 times m-reduction [i] based on digital (59, 92, 583)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 22, 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 (37, 70, 518)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 35, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(91−32, 91, 594)-Net in Base 16 — Constructive

(59, 91, 594)-net in base 16, using

1 times m-reduction [i] based on (59, 92, 594)-net in base 16, using
- (u,Â u+v)-construction [i] based on
  1. (10, 26, 80)-net in base 16, using
    - 1 times m-reduction [i] based on (10, 27, 80)-net in base 16, using
      - base change [i] based on digital (1, 18, 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 (33, 66, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 33, 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]

(91−32, 91, 3273)-Net over F₁₆ — Digital

Digital (59, 91, 3273)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹¹, 3273, F₁₆, 32) (dual of [3273, 3182, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁹¹, 4099, F₁₆, 32) (dual of [4099, 4008, 33]-code), using
  - 1 times truncation [i] based on linear OA(16⁹², 4100, F₁₆, 33) (dual of [4100, 4008, 34]-code), using
    - constructionÂ X applied to C_e(32) âŠ‚ C_e(30) [i] based on
      1. linear OA(16⁹¹, 4096, F₁₆, 33) (dual of [4096, 4005, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
      2. linear OA(16⁸⁸, 4096, F₁₆, 31) (dual of [4096, 4008, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
      3. linear OA(16¹, 4, F₁₆, 1) (dual of [4, 3, 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]

(91−32, 91, 3198001)-Net in Base 16 — Upper bound on s

There is no (59, 91, 3198002)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 37 576719 352328 525085 021728 557294 455760 533587 219169 422816 169460 904262 899580 221076 178195 365054 730146 304599 955231 > 16⁹¹ [i]