MinT - Best Known (115−40, 115, s)-Nets in Base 16

Best Known (115−40, 115, s)-Nets in Base 16

(115−40, 115, 587)-Net over F₁₆ — Constructive and digital

Digital (75, 115, 587)-net over F₁₆, using

1 times m-reduction [i] based on digital (75, 116, 587)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 26, 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 (49, 90, 522)-net over F₁₆, using
    - trace code for nets [i] based on digital (4, 45, 261)-net over F₂₅₆, using
      - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(115−40, 115, 643)-Net in Base 16 — Constructive

(75, 115, 643)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (15, 35, 129)-net in base 16, using
  - base change [i] based on digital (0, 20, 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 (40, 80, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 40, 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]

(115−40, 115, 4083)-Net over F₁₆ — Digital

Digital (75, 115, 4083)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹¹⁵, 4083, F₁₆, 40) (dual of [4083, 3968, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(16¹¹⁵, 4111, F₁₆, 40) (dual of [4111, 3996, 41]-code), using
  - constructionÂ X applied to C_e(39) âŠ‚ C_e(35) [i] based on
    1. linear OA(16¹¹², 4096, F₁₆, 40) (dual of [4096, 3984, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
    2. linear OA(16¹⁰⁰, 4096, F₁₆, 36) (dual of [4096, 3996, 37]-code), using an extension C_e(35) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [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]

(115−40, 115, 4644124)-Net in Base 16 — Upper bound on s

There is no (75, 115, 4644125)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 977137 116940 136118 260416 505329 922891 065866 528492 941380 332923 640914 369513 327118 245751 343590 791394 704253 215793 902361 302507 369462 397898 675001 > 16¹¹⁵ [i]