MinT - Best Known (61, 61+28, s)-Nets in Base 5

Best Known (61, 61+28, s)-Nets in Base 5

(61, 61+28, 252)-Net over F₅ — Constructive and digital

Digital (61, 89, 252)-net over F₅, using

13 times m-reduction [i] based on digital (61, 102, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 51, 126)-net over F₂₅, using
  - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
      - the Hermitian function field over F₂₅ [i]

(61, 61+28, 595)-Net over F₅ — Digital

Digital (61, 89, 595)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁸⁹, 595, F₅, 28) (dual of [595, 506, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁸⁹, 624, F₅, 28) (dual of [624, 535, 29]-code), using
  - the primitive BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â {−7,−6,…,20}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(61, 61+28, 41951)-Net in Base 5 — Upper bound on s

There is no (61, 89, 41952)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 161 591276 739571 866895 798720 924613 600503 871085 153999 164471 924225 > 5⁸⁹ [i]