MinT - Best Known (24, 24+27, s)-Nets in Base 25

Best Known (24, 24+27, s)-Nets in Base 25

(24, 24+27, 153)-Net over F₂₅ — Constructive and digital

Digital (24, 51, 153)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (1, 14, 27)-net over F₂₅, using
  - net from sequence [i] based on digital (1, 26)-sequence over F₂₅, using
    - Niederreiter sequence [i]
2. digital (10, 37, 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]

(24, 24+27, 280)-Net over F₂₅ — Digital

Digital (24, 51, 280)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(25⁵¹, 280, F₂₅, 2, 27) (dual of [(280, 2), 509, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25⁵¹, 313, F₂₅, 2, 27) (dual of [(313, 2), 575, 28]-NRT-code), using
  - 25¹ times duplication [i] based on linear OOA(25⁵⁰, 313, F₂₅, 2, 27) (dual of [(313, 2), 576, 28]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(25⁵⁰, 626, F₂₅, 27) (dual of [626, 576, 28]-code), using
      - constructionÂ X applied to C_e(26) âŠ‚ C_e(25) [i] based on
        linear OA(25⁵⁰, 625, F₂₅, 27) (dual of [625, 575, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
        linear OA(25⁴⁹, 625, F₂₅, 26) (dual of [625, 576, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
        linear OA(25⁰, 1, F₂₅, 0) (dual of [1, 1, 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]

(24, 24+27, 56212)-Net in Base 25 — Upper bound on s

There is no (24, 51, 56213)-net in base 25, because

1 times m-reduction [i] would yield (24, 50, 56213)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 7888 614182 801357 191039 438732 569973 292507 660392 723025 445137 901987 878425 > 25⁵⁰ [i]