MinT - Best Known (59−17, 59, s)-Nets in Base 16

Best Known (59−17, 59, s)-Nets in Base 16

(59−17, 59, 1285)-Net over F₁₆ — Constructive and digital

Digital (42, 59, 1285)-net over F₁₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 9, 257)-net over F₁₆, using
  - base reduction for projective spaces (embedding PG(4,256) in PG(8,16)) for nets [i] based on digital (0, 5, 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 (8, 16, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 8, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)
3. digital (17, 34, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 17, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(59−17, 59, 2048)-Net in Base 16 — Constructive

(42, 59, 2048)-net in base 16, using

16¹ times duplication [i] based on (41, 58, 2048)-net in base 16, using
- net defined by OOA [i] based on OOA(16⁵⁸, 2048, S₁₆, 17, 17), using
  - OOA 8-folding and stacking with additional row [i] based on OA(16⁵⁸, 16385, S₁₆, 17), using
    - discarding factors based on OA(16⁵⁸, 16386, S₁₆, 17), using
      - discarding parts of the base [i] based on linear OA(128³³, 16386, F₁₂₈, 17) (dual of [16386, 16353, 18]-code), using
        constructionÂ X applied to C_e(16) âŠ‚ C_e(15) [i] based on
        linear OA(128³³, 16384, F₁₂₈, 17) (dual of [16384, 16351, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
        linear OA(128³¹, 16384, F₁₂₈, 16) (dual of [16384, 16353, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
        linear OA(128⁰, 2, F₁₂₈, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(59−17, 59, 12500)-Net over F₁₆ — Digital

Digital (42, 59, 12500)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(59−17, 59, large)-Net in Base 16 — Upper bound on s

There is no (42, 59, large)-net in base 16, because

15 times m-reduction [i] would yield (42, 44, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]