MinT - Best Known (59−20, 59, s)-Nets in Base 16

Best Known (59−20, 59, s)-Nets in Base 16

(59−20, 59, 771)-Net over F₁₆ — Constructive and digital

Digital (39, 59, 771)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (9, 19, 257)-net over F₁₆, using
  - base reduction for projective spaces (embedding PG(9,256) in PG(18,16)) for nets [i] based on digital (0, 10, 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]
2. digital (20, 40, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 20, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(59−20, 59, 3810)-Net over F₁₆ — Digital

Digital (39, 59, 3810)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁵⁹, 3810, F₁₆, 20) (dual of [3810, 3751, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁵⁹, 4112, F₁₆, 20) (dual of [4112, 4053, 21]-code), using
  - constructionÂ X applied to C_e(19) âŠ‚ C_e(14) [i] based on
    1. linear OA(16⁵⁵, 4096, F₁₆, 20) (dual of [4096, 4041, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    2. linear OA(16⁴³, 4096, F₁₆, 15) (dual of [4096, 4053, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
    3. linear OA(16⁴, 16, F₁₆, 4) (dual of [16, 12, 5]-code or 16-arc in PG(3,16)), using
      - Reedâ€“Solomon code RS(12,16) [i]

(59−20, 59, 3838772)-Net in Base 16 — Upper bound on s

There is no (39, 59, 3838773)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 110428 130101 916509 150153 426328 155835 369309 455494 123523 045607 109277 626076 > 16⁵⁹ [i]