MinT - Best Known (57−19, 57, s)-Nets in Base 4

Best Known (57−19, 57, s)-Nets in Base 4

(57−19, 57, 195)-Net over F₄ — Constructive and digital

Digital (38, 57, 195)-net over F₄, using

trace code for nets [i] based on digital (0, 19, 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]

(57−19, 57, 219)-Net over F₄ — Digital

Digital (38, 57, 219)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁵⁷, 219, F₄, 19) (dual of [219, 162, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁵⁷, 255, F₄, 19) (dual of [255, 198, 20]-code), using
  - the primitive BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â {−5,−4,…,13}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(57−19, 57, 7698)-Net in Base 4 — Upper bound on s

There is no (38, 57, 7699)-net in base 4, because

1 times m-reduction [i] would yield (38, 56, 7699)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5197 422589 609300 345848 329163 007804 > 4⁵⁶ [i]