MinT - Best Known (42−17, 42, s)-Nets in Base 25

Best Known (42−17, 42, s)-Nets in Base 25

(42−17, 42, 252)-Net over F₂₅ — Constructive and digital

Digital (25, 42, 252)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (7, 15, 157)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25¹⁵, 157, F₂₅, 8, 8) (dual of [(157, 8), 1241, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(25¹⁵, 628, F₂₅, 8) (dual of [628, 613, 9]-code), using
      - constructionÂ XX applied to C₁Â =Â C([623,5]), C₂Â =Â C([0,6]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,5]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([623,6]) [i] based on
        linear OA(25¹³, 624, F₂₅, 7) (dual of [624, 611, 8]-code), using the primitive BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â {−1,0,…,5}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(25¹³, 624, F₂₅, 7) (dual of [624, 611, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(25¹⁵, 624, F₂₅, 8) (dual of [624, 609, 9]-code), using the primitive BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â {−1,0,…,6}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(25¹¹, 624, F₂₅, 6) (dual of [624, 613, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(25⁰, 2, F₂₅, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁰, s, F₂₅, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
        linear OA(25⁰, 2, F₂₅, 0) (dual of [2, 2, 1]-code) (see above)
2. digital (10, 27, 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]

(42−17, 42, 1332)-Net over F₂₅ — Digital

Digital (25, 42, 1332)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(42−17, 42, 2290449)-Net in Base 25 — Upper bound on s

There is no (25, 42, 2290450)-net in base 25, because

1 times m-reduction [i] would yield (25, 41, 2290450)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 2067 958172 738626 884434 414567 660987 369421 896425 448476 344961 > 25⁴¹ [i]

Best Known (42−17, 42, s)-Nets in Base 25

(42−17, 42, 252)-Net over F25 — Constructive and digital

(42−17, 42, 1332)-Net over F25 — Digital

(42−17, 42, 2290449)-Net in Base 25 — Upper bound on s

(42−17, 42, 252)-Net over F₂₅ — Constructive and digital

(42−17, 42, 1332)-Net over F₂₅ — Digital