MinT - Best Known (24−13, 24, s)-Nets in Base 25

Best Known (24−13, 24, s)-Nets in Base 25

(24−13, 24, 126)-Net over F₂₅ — Constructive and digital

Digital (11, 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]

(24−13, 24, 167)-Net over F₂₅ — Digital

Digital (11, 24, 167)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25²⁴, 167, F₂₅, 13) (dual of [167, 143, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(25²⁴, 208, F₂₅, 13) (dual of [208, 184, 14]-code), using
  - the BCH-code C(I) with length 208Â | 25²âˆ’1, defining interval IÂ =Â {1,4,7,…,37}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(24−13, 24, 28493)-Net in Base 25 — Upper bound on s

There is no (11, 24, 28494)-net in base 25, because

1 times m-reduction [i] would yield (11, 23, 28494)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 142 136933 401184 563550 003437 906849 > 25²³ [i]