MinT - Best Known (41, 79, s)-Nets in Base 25

Best Known (41, 79, s)-Nets in Base 25

(41, 79, 288)-Net over F₂₅ — Constructive and digital

Digital (41, 79, 288)-net over F₂₅, using

net from sequence [i] based on digital (41, 287)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 41 and N(F) ≥ 288, using
  - a function field by GarcÃa and Quoos [i]

(41, 79, 619)-Net over F₂₅ — Digital

Digital (41, 79, 619)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁷⁹, 619, F₂₅, 38) (dual of [619, 540, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(25⁷⁹, 648, F₂₅, 38) (dual of [648, 569, 39]-code), using
  - constructionÂ X applied to C_e(37) âŠ‚ C_e(29) [i] based on
    1. linear OA(25⁷², 625, F₂₅, 38) (dual of [625, 553, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
    2. linear OA(25⁵⁶, 625, F₂₅, 30) (dual of [625, 569, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
    3. linear OA(25⁷, 23, F₂₅, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,25)), using
      - discarding factors / shortening the dual code based on linear OA(25⁷, 25, F₂₅, 7) (dual of [25, 18, 8]-code or 25-arc in PG(6,25)), using
        Reedâ€“Solomon code RS(18,25) [i]

(41, 79, 214522)-Net in Base 25 — Upper bound on s

There is no (41, 79, 214523)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 273 715232 256063 795117 628731 966461 903458 120298 250860 312487 519721 597966 760117 169578 928823 059398 215547 751777 874585 > 25⁷⁹ [i]