MinT - Best Known (69−34, 69, s)-Nets in Base 25

Best Known (69−34, 69, s)-Nets in Base 25

(69−34, 69, 208)-Net over F₂₅ — Constructive and digital

Digital (35, 69, 208)-net over F₂₅, using

net from sequence [i] based on digital (35, 207)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 35 and N(F) ≥ 208, using
  - a function field by GarcÃa and Quoos [i]

(69−34, 69, 484)-Net over F₂₅ — Digital

Digital (35, 69, 484)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁶⁹, 484, F₂₅, 34) (dual of [484, 415, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(25⁶⁹, 642, F₂₅, 34) (dual of [642, 573, 35]-code), using
  - constructionÂ X applied to C_e(33) âŠ‚ C_e(27) [i] based on
    1. linear OA(25⁶⁴, 625, F₂₅, 34) (dual of [625, 561, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    2. linear OA(25⁵², 625, F₂₅, 28) (dual of [625, 573, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    3. linear OA(25⁵, 17, F₂₅, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,25)), using
      - discarding factors / shortening the dual code based on linear OA(25⁵, 25, F₂₅, 5) (dual of [25, 20, 6]-code or 25-arc in PG(4,25)), using
        Reedâ€“Solomon code RS(20,25) [i]

(69−34, 69, 141155)-Net in Base 25 — Upper bound on s

There is no (35, 69, 141156)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 2 869900 797939 583684 394003 368639 805326 103795 875801 680123 828338 425463 582107 133320 477641 251139 165025 > 25⁶⁹ [i]