MinT - Best Known (45−13, 45, s)-Nets in Base 4

Best Known (45−13, 45, s)-Nets in Base 4

(45−13, 45, 240)-Net over F₄ — Constructive and digital

Digital (32, 45, 240)-net over F₄, using

t-expansion [i] based on digital (31, 45, 240)-net over F₄, using
- trace code for nets [i] based on digital (1, 15, 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]

(45−13, 45, 341)-Net over F₄ — Digital

Digital (32, 45, 341)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁴⁵, 341, F₄, 13) (dual of [341, 296, 14]-code), using
- the BCH-code C(I) with length 341Â | 4⁵âˆ’1, defining interval IÂ =Â {1,52,103,…,−69}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(45−13, 45, 25949)-Net in Base 4 — Upper bound on s

There is no (32, 45, 25950)-net in base 4, because

1 times m-reduction [i] would yield (32, 44, 25950)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 309 507374 681256 357967 911796 > 4⁴⁴ [i]