MinT - Best Known (53, 81, s)-Nets in Base 16

Best Known (53, 81, s)-Nets in Base 16

(53, 81, 583)-Net over F₁₆ — Constructive and digital

Digital (53, 81, 583)-net over F₁₆, using

1 times m-reduction [i] based on digital (53, 82, 583)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 20, 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 (33, 62, 518)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 31, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(53, 81, 643)-Net in Base 16 — Constructive

(53, 81, 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 (28, 56, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 28, 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]

(53, 81, 3553)-Net over F₁₆ — Digital

Digital (53, 81, 3553)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸¹, 3553, F₁₆, 28) (dual of [3553, 3472, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁸¹, 4107, F₁₆, 28) (dual of [4107, 4026, 29]-code), using
  - constructionÂ X applied to C_e(27) âŠ‚ C_e(24) [i] based on
    1. linear OA(16⁷⁹, 4096, F₁₆, 28) (dual of [4096, 4017, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    2. linear OA(16⁷⁰, 4096, F₁₆, 25) (dual of [4096, 4026, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [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]

(53, 81, 3733015)-Net in Base 16 — Upper bound on s

There is no (53, 81, 3733016)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 34 175848 089262 353158 962565 962569 333020 047874 180875 058351 155407 318284 675899 941992 801647 607839 562086 > 16⁸¹ [i]