MinT - Best Known (54, 78, s)-Nets in Base 5

Best Known (54, 78, s)-Nets in Base 5

(54, 78, 252)-Net over F₅ — Constructive and digital

Digital (54, 78, 252)-net over F₅, using

10 times m-reduction [i] based on digital (54, 88, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 44, 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]

(54, 78, 618)-Net over F₅ — Digital

Digital (54, 78, 618)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁷⁸, 618, F₅, 24) (dual of [618, 540, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁷⁸, 634, F₅, 24) (dual of [634, 556, 25]-code), using
  - constructionÂ X applied to C_e(23) âŠ‚ C_e(21) [i] based on
    1. linear OA(5⁷⁷, 625, F₅, 24) (dual of [625, 548, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]
    2. linear OA(5⁶⁹, 625, F₅, 22) (dual of [625, 556, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [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]

(54, 78, 46187)-Net in Base 5 — Upper bound on s

There is no (54, 78, 46188)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3 308961 738872 686924 472824 754816 361094 225111 536082 952961 > 5⁷⁸ [i]