MinT - Best Known (83−40, 83, s)-Nets in Base 25

Best Known (83−40, 83, s)-Nets in Base 25

(83−40, 83, 288)-Net over F₂₅ — Constructive and digital

Digital (43, 83, 288)-net over F₂₅, using

t-expansion [i] based on digital (41, 83, 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]

(83−40, 83, 633)-Net over F₂₅ — Digital

Digital (43, 83, 633)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁸³, 633, F₂₅, 40) (dual of [633, 550, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(25⁸³, 648, F₂₅, 40) (dual of [648, 565, 41]-code), using
  - constructionÂ X applied to C_e(39) âŠ‚ C_e(31) [i] based on
    1. linear OA(25⁷⁶, 625, F₂₅, 40) (dual of [625, 549, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
    2. linear OA(25⁶⁰, 625, F₂₅, 32) (dual of [625, 565, 33]-code), using an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [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]

(83−40, 83, 219041)-Net in Base 25 — Upper bound on s

There is no (43, 83, 219042)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 106 919737 844865 484410 915847 917869 673742 357998 702463 559555 899688 665533 774933 731531 977309 744589 756775 331173 216218 708545 > 25⁸³ [i]