MinT - Best Known (97−31, 97, s)-Nets in Base 5

Best Known (97−31, 97, s)-Nets in Base 5

(97−31, 97, 258)-Net over F₅ — Constructive and digital

Digital (66, 97, 258)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (0, 15, 6)-net over F₅, using
  - net from sequence [i] based on digital (0, 5)-sequence over F₅, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 0 and N(F) ≥ 6, using
      - the rational function field F₅(x) [i]
    - Niederreiter sequence [i]
2. digital (51, 82, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 41, 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]

(97−31, 97, 582)-Net over F₅ — Digital

Digital (66, 97, 582)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁹⁷, 582, F₅, 31) (dual of [582, 485, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁹⁷, 624, F₅, 31) (dual of [624, 527, 32]-code), using
  - the primitive BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â {136,137,…,166}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]

(97−31, 97, 47754)-Net in Base 5 — Upper bound on s

There is no (66, 97, 47755)-net in base 5, because

1 times m-reduction [i] would yield (66, 96, 47755)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 12 622609 808186 464545 822045 122636 448675 099781 115536 190086 654421 449525 > 5⁹⁶ [i]