MinT - Best Known (59, 59+31, s)-Nets in Base 16

Best Known (59, 59+31, s)-Nets in Base 16

(59, 59+31, 596)-Net over F₁₆ — Constructive and digital

Digital (59, 90, 596)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (13, 28, 82)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 7, 17)-net over F₁₆, using
      - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
        the rational function field F₁₆(x) [i]
        Niederreiter sequence [i]
    2. digital (6, 21, 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 (31, 62, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 31, 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]

(59, 59+31, 664)-Net in Base 16 — Constructive

(59, 90, 664)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (13, 28, 150)-net in base 16, using
  - base change [i] based on digital (1, 16, 150)-net over F₁₂₈, using
    - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]
2. digital (31, 62, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 31, 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]

(59, 59+31, 3843)-Net over F₁₆ — Digital

Digital (59, 90, 3843)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹⁰, 3843, F₁₆, 31) (dual of [3843, 3753, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁹⁰, 4107, F₁₆, 31) (dual of [4107, 4017, 32]-code), using
  - constructionÂ X applied to C_e(30) âŠ‚ C_e(27) [i] based on
    1. 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]
    2. 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]
    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]

(59, 59+31, 5972003)-Net in Base 16 — Upper bound on s

There is no (59, 90, 5972004)-net in base 16, because

1 times m-reduction [i] would yield (59, 89, 5972004)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 146784 213591 428213 333535 790048 003368 770898 742636 429804 526349 299215 890237 984377 595412 875158 542894 512059 552776 > 16⁸⁹ [i]