MinT - Best Known (29−10, 29, s)-Nets in Base 4

Best Known (29−10, 29, s)-Nets in Base 4

(29−10, 29, 90)-Net over F₄ — Constructive and digital

Digital (19, 29, 90)-net over F₄, using

1 times m-reduction [i] based on digital (19, 30, 90)-net over F₄, using
- trace code for nets [i] based on digital (4, 15, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]

(29−10, 29, 155)-Net over F₄ — Digital

Digital (19, 29, 155)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁹, 155, F₄, 10) (dual of [155, 126, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁹, 255, F₄, 10) (dual of [255, 226, 11]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(29−10, 29, 2691)-Net in Base 4 — Upper bound on s

There is no (19, 29, 2692)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 288240 062556 611968 > 4²⁹ [i]