MinT - Best Known (45, 45+21, s)-Nets in Base 5

Best Known (45, 45+21, s)-Nets in Base 5

(45, 45+21, 252)-Net over F₅ — Constructive and digital

Digital (45, 66, 252)-net over F₅, using

4 times m-reduction [i] based on digital (45, 70, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 35, 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]

(45, 45+21, 476)-Net over F₅ — Digital

Digital (45, 66, 476)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁶⁶, 476, F₅, 21) (dual of [476, 410, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁶⁶, 630, F₅, 21) (dual of [630, 564, 22]-code), using
  - constructionÂ X applied to C_e(20) âŠ‚ C_e(18) [i] based on
    1. linear OA(5⁶⁵, 625, F₅, 21) (dual of [625, 560, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
    2. linear OA(5⁶¹, 625, F₅, 19) (dual of [625, 564, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [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]

(45, 45+21, 39549)-Net in Base 5 — Upper bound on s

There is no (45, 66, 39550)-net in base 5, because

1 times m-reduction [i] would yield (45, 65, 39550)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2710 630389 845534 185646 800442 392458 568177 490561 > 5⁶⁵ [i]