MinT - Best Known (67−15, 67, s)-Nets in Base 4

Best Known (67−15, 67, s)-Nets in Base 4

(67−15, 67, 1033)-Net over F₄ — Constructive and digital

Digital (52, 67, 1033)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 7, 5)-net over F₄, using
  - net from sequence [i] based on digital (0, 4)-sequence over F₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 0 and N(F) ≥ 5, using
      - the rational function field F₄(x) [i]
    - Niederreiter sequence [i]
2. digital (45, 60, 1028)-net over F₄, 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, 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]

(67−15, 67, 2143)-Net over F₄ — Digital

Digital (52, 67, 2143)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁶⁷, 2143, F₄, 15) (dual of [2143, 2076, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁶⁷, 4096, F₄, 15) (dual of [4096, 4029, 16]-code), using
  - an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(67−15, 67, 535006)-Net in Base 4 — Upper bound on s

There is no (52, 67, 535007)-net in base 4, because

1 times m-reduction [i] would yield (52, 66, 535007)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5444 585249 374089 931522 097069 264865 075820 > 4⁶⁶ [i]