MinT - Best Known (76, 117, s)-Nets in Base 16

Best Known (76, 117, s)-Nets in Base 16

(76, 117, 587)-Net over F₁₆ — Constructive and digital

Digital (76, 117, 587)-net over F₁₆, using

16¹ times duplication [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]

(76, 117, 643)-Net in Base 16 — Constructive

(76, 117, 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 (41, 82, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 41, 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]

(76, 117, 3895)-Net over F₁₆ — Digital

Digital (76, 117, 3895)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹¹⁷, 3895, F₁₆, 41) (dual of [3895, 3778, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(16¹¹⁷, 4107, F₁₆, 41) (dual of [4107, 3990, 42]-code), using
  - constructionÂ X applied to C_e(40) âŠ‚ C_e(37) [i] based on
    1. linear OA(16¹¹⁵, 4096, F₁₆, 41) (dual of [4096, 3981, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
    2. linear OA(16¹⁰⁶, 4096, F₁₆, 38) (dual of [4096, 3990, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [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]

(76, 117, 5334699)-Net in Base 16 — Upper bound on s

There is no (76, 117, 5334700)-net in base 16, because

1 times m-reduction [i] would yield (76, 116, 5334700)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 47 634129 796249 379030 520188 235163 812845 922422 629763 425000 639916 342272 118149 902354 488855 056266 753605 148105 368694 778585 010472 138889 011999 106876 > 16¹¹⁶ [i]