MinT - Best Known (95−31, 95, s)-Nets in Base 5

Best Known (95−31, 95, s)-Nets in Base 5

(95−31, 95, 252)-Net over F₅ — Constructive and digital

Digital (64, 95, 252)-net over F₅, using

13 times m-reduction [i] based on digital (64, 108, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 54, 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]

(95−31, 95, 519)-Net over F₅ — Digital

Digital (64, 95, 519)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁹⁵, 519, F₅, 31) (dual of [519, 424, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁹⁵, 624, F₅, 31) (dual of [624, 529, 32]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [0,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]

(95−31, 95, 38529)-Net in Base 5 — Upper bound on s

There is no (64, 95, 38530)-net in base 5, because

1 times m-reduction [i] would yield (64, 94, 38530)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 504896 662682 416567 363427 563885 011567 175645 092111 009346 904768 423465 > 5⁹⁴ [i]