MinT - Best Known (14, 21, s)-Nets in Base 4

Best Known (14, 21, s)-Nets in Base 4

(14, 21, 195)-Net over F₄ — Constructive and digital

Digital (14, 21, 195)-net over F₄, using

trace code for nets [i] based on digital (0, 7, 65)-net over F₆₄, using
- net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
    - the rational function field F₆₄(x) [i]
  - Niederreiter sequence [i]

(14, 21, 219)-Net over F₄ — Digital

Digital (14, 21, 219)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²¹, 219, F₄, 7) (dual of [219, 198, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(4²¹, 255, F₄, 7) (dual of [255, 234, 8]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(14, 21, 6249)-Net in Base 4 — Upper bound on s

There is no (14, 21, 6250)-net in base 4, because

1 times m-reduction [i] would yield (14, 20, 6250)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 099687 696876 > 4²⁰ [i]