MinT - Best Known (66−20, 66, s)-Nets in Base 5

Best Known (66−20, 66, s)-Nets in Base 5

(66−20, 66, 252)-Net over F₅ — Constructive and digital

Digital (46, 66, 252)-net over F₅, using

6 times m-reduction [i] based on digital (46, 72, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 36, 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]

(66−20, 66, 619)-Net over F₅ — Digital

Digital (46, 66, 619)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁶⁶, 619, F₅, 20) (dual of [619, 553, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁶⁶, 634, F₅, 20) (dual of [634, 568, 21]-code), using
  - constructionÂ X applied to C_e(20) âŠ‚ C_e(17) [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(5⁵⁷, 625, F₅, 18) (dual of [625, 568, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
    3. linear OA(5¹, 9, F₅, 1) (dual of [9, 8, 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]

(66−20, 66, 46457)-Net in Base 5 — Upper bound on s

There is no (46, 66, 46458)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 13554 898149 741580 454279 783556 941544 893710 247745 > 5⁶⁶ [i]