MinT - Best Known (74, 114, s)-Nets in Base 16

Best Known (74, 114, s)-Nets in Base 16

(74, 114, 587)-Net over F₁₆ — Constructive and digital

Digital (74, 114, 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 (48, 88, 522)-net over F₁₆, using
  - trace code for nets [i] based on digital (4, 44, 261)-net over F₂₅₆, using
    - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(74, 114, 612)-Net in Base 16 — Constructive

(74, 114, 612)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (14, 34, 98)-net in base 16, using
  - 1 times m-reduction [i] based on (14, 35, 98)-net in base 16, using
    - base change [i] based on digital (7, 28, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [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]

(74, 114, 3794)-Net over F₁₆ — Digital

Digital (74, 114, 3794)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹¹⁴, 3794, F₁₆, 40) (dual of [3794, 3680, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(16¹¹⁴, 4107, F₁₆, 40) (dual of [4107, 3993, 41]-code), using
  - constructionÂ X applied to C_e(39) âŠ‚ C_e(36) [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(16¹⁰³, 4096, F₁₆, 37) (dual of [4096, 3993, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [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]

(74, 114, 4042943)-Net in Base 16 — Upper bound on s

There is no (74, 114, 4042944)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 186070 855891 022456 190880 470384 593831 308353 811048 325482 766233 595959 920826 041192 469080 431188 779250 196630 044639 751306 087207 564897 900052 230701 > 16¹¹⁴ [i]