MinT - Best Known (5, 5+4, s)-Nets in Base 16

Best Known (5, 5+4, s)-Nets in Base 16

(5, 5+4, 514)-Net over F₁₆ — Constructive and digital

Digital (5, 9, 514)-net over F₁₆, using

1 times m-reduction [i] based on digital (5, 10, 514)-net over F₁₆, using
- trace code 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]

(5, 5+4, 819)-Net over F₁₆ — Digital

Digital (5, 9, 819)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹, 819, F₁₆, 4) (dual of [819, 810, 5]-code), using
- the BCH-code C(I) with length 819Â | 16³âˆ’1, defining interval IÂ =Â {1,6,11,16}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

(5, 5+4, 2016)-Net in Base 16 — Constructive

(5, 9, 2016)-net in base 16, using

base change [i] based on digital (2, 6, 2016)-net over F₆₄, using
- net defined by OOA [i] based on linear OOA(64⁶, 2016, F₆₄, 4, 4) (dual of [(2016, 4), 8058, 5]-NRT-code), using
  - appending kth column [i] based on linear OOA(64⁶, 2016, F₆₄, 3, 4) (dual of [(2016, 3), 6042, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(64⁶, 4032, F₆₄, 4) (dual of [4032, 4026, 5]-code), using
      - 1 times truncation [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
        a code by Danev and Olsson [i]

(5, 5+4, 24714)-Net in Base 16 — Upper bound on s

There is no (5, 9, 24715)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 68722 034701 > 16⁹ [i]