MinT - Best Known (55, 84, s)-Nets in Base 16

Best Known (55, 84, s)-Nets in Base 16

(55, 84, 585)-Net over F₁₆ — Constructive and digital

Digital (55, 84, 585)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (12, 26, 71)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (2, 9, 33)-net over F₁₆, using
      - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
        a function field by SÃ©mirat [i]
    2. digital (3, 17, 38)-net over F₁₆, using
      - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
2. digital (29, 58, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 29, 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]

(55, 84, 643)-Net in Base 16 — Constructive

(55, 84, 643)-net in base 16, using

16¹ times duplication [i] based on (54, 83, 643)-net in base 16, using
- (u,Â u+v)-construction [i] based on
  1. (11, 25, 129)-net in base 16, using
    - base change [i] based on (6, 20, 129)-net in base 32, using
      - 1 times m-reduction [i] based on (6, 21, 129)-net in base 32, using
        base change [i] based on digital (0, 15, 129)-net over F₁₂₈, using
        net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
  2. digital (29, 58, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 29, 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]

(55, 84, 3649)-Net over F₁₆ — Digital

Digital (55, 84, 3649)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸⁴, 3649, F₁₆, 29) (dual of [3649, 3565, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁸⁴, 4107, F₁₆, 29) (dual of [4107, 4023, 30]-code), using
  - constructionÂ X applied to C_e(28) âŠ‚ C_e(25) [i] based on
    1. linear OA(16⁸², 4096, F₁₆, 29) (dual of [4096, 4014, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    2. linear OA(16⁷³, 4096, F₁₆, 26) (dual of [4096, 4023, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    3. linear OA(16², 11, F₁₆, 2) (dual of [11, 9, 3]-code or 11-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]

(55, 84, 5547243)-Net in Base 16 — Upper bound on s

There is no (55, 84, 5547244)-net in base 16, because

1 times m-reduction [i] would yield (55, 83, 5547244)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 8749 015569 382614 623244 608092 515577 187137 492007 970608 506809 433208 410905 448889 927977 604233 294429 614616 > 16⁸³ [i]