MinT - Best Known (24−12, 24, s)-Nets in Base 25

Best Known (24−12, 24, s)-Nets in Base 25

(24−12, 24, 126)-Net over F₂₅ — Constructive and digital

Digital (12, 24, 126)-net over F₂₅, using

t-expansion [i] based on digital (10, 24, 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]

(24−12, 24, 315)-Net over F₂₅ — Digital

Digital (12, 24, 315)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(25²⁴, 315, F₂₅, 2, 12) (dual of [(315, 2), 606, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25²⁴, 630, F₂₅, 12) (dual of [630, 606, 13]-code), using
  - constructionÂ X applied to C_e(11) âŠ‚ C_e(9) [i] based on
    1. linear OA(25²³, 625, F₂₅, 12) (dual of [625, 602, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
    2. linear OA(25¹⁹, 625, F₂₅, 10) (dual of [625, 606, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
    3. linear OA(25¹, 5, F₂₅, 1) (dual of [5, 4, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(25¹, s, F₂₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(24−12, 24, 48724)-Net in Base 25 — Upper bound on s

There is no (12, 24, 48725)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 3552 982770 838550 931038 241164 555281 > 25²⁴ [i]