MinT - Best Known (51, 51+27, s)-Nets in Base 16

Best Known (51, 51+27, s)-Nets in Base 16

(51, 51+27, 585)-Net over F₁₆ — Constructive and digital

Digital (51, 78, 585)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (11, 24, 71)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (2, 8, 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, 16, 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 (27, 54, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 27, 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]

(51, 51+27, 643)-Net in Base 16 — Constructive

(51, 78, 643)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (11, 24, 129)-net in base 16, using
  - base change [i] based on (3, 16, 129)-net in base 64, using
    - 5 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
      - base change [i] based on digital (0, 18, 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 (27, 54, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 27, 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]

(51, 51+27, 3456)-Net over F₁₆ — Digital

Digital (51, 78, 3456)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁷⁸, 3456, F₁₆, 27) (dual of [3456, 3378, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁷⁸, 4107, F₁₆, 27) (dual of [4107, 4029, 28]-code), using
  - constructionÂ X applied to C_e(26) âŠ‚ C_e(23) [i] based on
    1. linear OA(16⁷⁶, 4096, F₁₆, 27) (dual of [4096, 4020, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
    2. linear OA(16⁶⁷, 4096, F₁₆, 24) (dual of [4096, 4029, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [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]

(51, 51+27, 5121646)-Net in Base 16 — Upper bound on s

There is no (51, 78, 5121647)-net in base 16, because

1 times m-reduction [i] would yield (51, 77, 5121647)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 521 481576 383099 834809 351075 671137 979219 904864 702322 957743 192544 802158 081119 604033 447177 671066 > 16⁷⁷ [i]