MinT - Best Known (56, 81, s)-Nets in Base 5

Best Known (56, 81, s)-Nets in Base 5

(56, 81, 252)-Net over F₅ — Constructive and digital

Digital (56, 81, 252)-net over F₅, using

11 times m-reduction [i] based on digital (56, 92, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 46, 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]

(56, 81, 621)-Net over F₅ — Digital

Digital (56, 81, 621)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁸¹, 621, F₅, 25) (dual of [621, 540, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁸¹, 624, F₅, 25) (dual of [624, 543, 26]-code), using
  - the primitive BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â {−7,−6,…,17}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(56, 81, 60400)-Net in Base 5 — Upper bound on s

There is no (56, 81, 60401)-net in base 5, because

1 times m-reduction [i] would yield (56, 80, 60401)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 82 726716 624466 397208 658014 617262 308435 921061 172839 737425 > 5⁸⁰ [i]