MinT - Best Known (28, 43, s)-Nets in Base 16

Best Known (28, 43, s)-Nets in Base 16

(28, 43, 771)-Net over F₁₆ — Constructive and digital

Digital (28, 43, 771)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 13, 257)-net over F₁₆, using
  - base reduction for projective spaces (embedding PG(6,256) in PG(12,16)) for nets [i] based on digital (0, 7, 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 (15, 30, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 15, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(28, 43, 2928)-Net over F₁₆ — Digital

Digital (28, 43, 2928)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁴³, 2928, F₁₆, 15) (dual of [2928, 2885, 16]-code), using
- discarding factors / shortening the dual code based on 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]

(28, 43, 3780479)-Net in Base 16 — Upper bound on s

There is no (28, 43, 3780480)-net in base 16, because

1 times m-reduction [i] would yield (28, 42, 3780480)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 374 144769 782293 084287 863104 224078 483244 639964 954401 > 16⁴² [i]