MinT - Best Known (65, 96, s)-Nets in Base 5

Best Known (65, 96, s)-Nets in Base 5

(65, 96, 252)-Net over F₅ — Constructive and digital

Digital (65, 96, 252)-net over F₅, using

14 times m-reduction [i] based on digital (65, 110, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 55, 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]

(65, 96, 550)-Net over F₅ — Digital

Digital (65, 96, 550)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁹⁶, 550, F₅, 31) (dual of [550, 454, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁹⁶, 630, F₅, 31) (dual of [630, 534, 32]-code), using
  - constructionÂ X applied to C_e(30) âŠ‚ C_e(28) [i] based on
    1. linear OA(5⁹⁵, 625, F₅, 31) (dual of [625, 530, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    2. linear OA(5⁹¹, 625, F₅, 29) (dual of [625, 534, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    3. linear OA(5¹, 5, F₅, 1) (dual of [5, 4, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(5¹, s, F₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(65, 96, 42895)-Net in Base 5 — Upper bound on s

There is no (65, 96, 42896)-net in base 5, because

1 times m-reduction [i] would yield (65, 95, 42896)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2 525133 872013 975779 301450 757049 193076 705439 417083 284383 929272 198721 > 5⁹⁵ [i]