MinT - Best Known (30−9, 30, s)-Nets in Base 4

Best Known (30−9, 30, s)-Nets in Base 4

(30−9, 30, 240)-Net over F₄ — Constructive and digital

Digital (21, 30, 240)-net over F₄, using

trace code for nets [i] based on digital (1, 10, 80)-net over F₆₄, using
- net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
    - a function field by SÃ©mirat [i]

(30−9, 30, 341)-Net over F₄ — Digital

Digital (21, 30, 341)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4³⁰, 341, F₄, 9) (dual of [341, 311, 10]-code), using
- the BCH-code C(I) with length 341Â | 4⁵âˆ’1, defining interval IÂ =Â {1,52,103,…,68}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(30−9, 30, 17092)-Net in Base 4 — Upper bound on s

There is no (21, 30, 17093)-net in base 4, because

1 times m-reduction [i] would yield (21, 29, 17093)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 288293 505265 488481 > 4²⁹ [i]