MinT - Best Known (51, 51+23, s)-Nets in Base 5

Best Known (51, 51+23, s)-Nets in Base 5

(51, 51+23, 252)-Net over F₅ — Constructive and digital

Digital (51, 74, 252)-net over F₅, using

8 times m-reduction [i] based on 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]

(51, 51+23, 570)-Net over F₅ — Digital

Digital (51, 74, 570)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁷⁴, 570, F₅, 23) (dual of [570, 496, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁷⁴, 635, F₅, 23) (dual of [635, 561, 24]-code), using
  - constructionÂ X applied to C([0,11]) âŠ‚ C([0,10]) [i] based on
    1. linear OA(5⁷³, 626, F₅, 23) (dual of [626, 553, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 626Â | 5⁸âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
    2. linear OA(5⁶⁵, 626, F₅, 21) (dual of [626, 561, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 626Â | 5⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [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]

(51, 51+23, 53396)-Net in Base 5 — Upper bound on s

There is no (51, 74, 53397)-net in base 5, because

1 times m-reduction [i] would yield (51, 73, 53397)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1058 968153 062118 020697 919759 983107 982636 881537 685869 > 5⁷³ [i]