MinT - Best Known (63−21, 63, s)-Nets in Base 4

Best Known (63−21, 63, s)-Nets in Base 4

(63−21, 63, 195)-Net over F₄ — Constructive and digital

Digital (42, 63, 195)-net over F₄, using

trace code for nets [i] based on digital (0, 21, 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]

(63−21, 63, 231)-Net over F₄ — Digital

Digital (42, 63, 231)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁶³, 231, F₄, 21) (dual of [231, 168, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁶³, 255, F₄, 21) (dual of [255, 192, 22]-code), using
  - the primitive BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â {−2,−1,…,18}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(63−21, 63, 8150)-Net in Base 4 — Upper bound on s

There is no (42, 63, 8151)-net in base 4, because

1 times m-reduction [i] would yield (42, 62, 8151)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 21 268093 290613 653703 582155 561671 787886 > 4⁶² [i]