MinT - Best Known (96−34, 96, s)-Nets in Base 16

Best Known (96−34, 96, s)-Nets in Base 16

(96−34, 96, 583)-Net over F₁₆ — Constructive and digital

Digital (62, 96, 583)-net over F₁₆, using

1 times m-reduction [i] based on digital (62, 97, 583)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 23, 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 (39, 74, 518)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 37, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(96−34, 96, 594)-Net in Base 16 — Constructive

(62, 96, 594)-net in base 16, using

1 times m-reduction [i] based on (62, 97, 594)-net in base 16, using
- (u,Â u+v)-construction [i] based on
  1. (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 (35, 70, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 35, 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]

(96−34, 96, 3186)-Net over F₁₆ — Digital

Digital (62, 96, 3186)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹⁶, 3186, F₁₆, 34) (dual of [3186, 3090, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁹⁶, 4104, F₁₆, 34) (dual of [4104, 4008, 35]-code), using
  - constructionÂ X applied to C_e(33) âŠ‚ C_e(30) [i] based on
    1. linear OA(16⁹⁴, 4096, F₁₆, 34) (dual of [4096, 4002, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [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(16², 8, F₁₆, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,16)), using
      - discarding factors / shortening the dual code based on linear OA(16², 16, F₁₆, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
        Reedâ€“Solomon code RS(14,16) [i]

(96−34, 96, 3017095)-Net in Base 16 — Upper bound on s

There is no (62, 96, 3017096)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 39 402196 518400 658316 913622 487857 698787 860044 468689 668751 182481 685840 911347 538470 849111 157520 486936 612793 651782 112456 > 16⁹⁶ [i]