MinT - Best Known (96−25, 96, s)-Nets in Base 4

Best Known (96−25, 96, s)-Nets in Base 4

(96−25, 96, 531)-Net over F₄ — Constructive and digital

Digital (71, 96, 531)-net over F₄, using

trace code for nets [i] based on digital (7, 32, 177)-net over F₆₄, using
- net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
    - a function field by GarcÃa and Quoos [i]

(96−25, 96, 947)-Net over F₄ — Digital

Digital (71, 96, 947)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁹⁶, 947, F₄, 25) (dual of [947, 851, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁹⁶, 1023, F₄, 25) (dual of [1023, 927, 26]-code), using
  - the primitive BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â {−6,−5,…,18}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(96−25, 96, 102922)-Net in Base 4 — Upper bound on s

There is no (71, 96, 102923)-net in base 4, because

1 times m-reduction [i] would yield (71, 95, 102923)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1569 453043 379531 235483 346210 591522 002971 590106 628861 213088 > 4⁹⁵ [i]