MinT - Best Known (82−28, 82, s)-Nets in Base 16

Best Known (82−28, 82, s)-Nets in Base 16

(82−28, 82, 585)-Net over F₁₆ — Constructive and digital

Digital (54, 82, 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 (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]

(82−28, 82, 643)-Net in Base 16 — Constructive

(54, 82, 643)-net in base 16, using

1 times m-reduction [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]

(82−28, 82, 3954)-Net over F₁₆ — Digital

Digital (54, 82, 3954)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸², 3954, F₁₆, 28) (dual of [3954, 3872, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁸², 4111, F₁₆, 28) (dual of [4111, 4029, 29]-code), using
  - constructionÂ X applied to C_e(27) âŠ‚ C_e(23) [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(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(16³, 15, F₁₆, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,16) or 15-cap in PG(2,16)), using
      - discarding factors / shortening the dual code based on linear OA(16³, 16, F₁₆, 3) (dual of [16, 13, 4]-code or 16-arc in PG(2,16) or 16-cap in PG(2,16)), using
        Reedâ€“Solomon code RS(13,16) [i]

(82−28, 82, 4550598)-Net in Base 16 — Upper bound on s

There is no (54, 82, 4550599)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 546 813395 541618 488184 038107 625955 948827 345479 299305 019254 665715 758778 656002 301256 836583 910115 211416 > 16⁸² [i]