MinT - Best Known (51, 51+22, s)-Nets in Base 5

Best Known (51, 51+22, s)-Nets in Base 5

(51, 51+22, 252)-Net over F₅ — Constructive and digital

Digital (51, 73, 252)-net over F₅, using

9 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+22, 650)-Net over F₅ — Digital

Digital (51, 73, 650)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁷³, 650, F₅, 22) (dual of [650, 577, 23]-code), using
- 17 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 4 times 0, 1, 9 times 0) [i] based on linear OA(5⁶⁹, 629, F₅, 22) (dual of [629, 560, 23]-code), using
  - constructionÂ X applied to C_e(21) âŠ‚ C_e(20) [i] based on
    1. 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]
    2. 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]
    3. linear OA(5⁰, 4, F₅, 0) (dual of [4, 4, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(5⁰, s, F₅, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(51, 51+22, 53396)-Net in Base 5 — Upper bound on s

There is no (51, 73, 53397)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1058 968153 062118 020697 919759 983107 982636 881537 685869 > 5⁷³ [i]