MinT - Best Known (49−23, 49, s)-Nets in Base 25

Best Known (49−23, 49, s)-Nets in Base 25

(49−23, 49, 200)-Net over F₂₅ — Constructive and digital

Digital (26, 49, 200)-net over F₂₅, using

t-expansion [i] based on digital (25, 49, 200)-net over F₂₅, using
- net from sequence [i] based on digital (25, 199)-sequence over F₂₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 25 and N(F) ≥ 200, using
    - a function field by GarcÃa and Quoos [i]

(49−23, 49, 557)-Net over F₂₅ — Digital

Digital (26, 49, 557)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁴⁹, 557, F₂₅, 23) (dual of [557, 508, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(25⁴⁹, 639, F₂₅, 23) (dual of [639, 590, 24]-code), using
  - constructionÂ X applied to C_e(22) âŠ‚ C_e(17) [i] based on
    1. linear OA(25⁴⁵, 625, F₂₅, 23) (dual of [625, 580, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    2. linear OA(25³⁵, 625, F₂₅, 18) (dual of [625, 590, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
    3. linear OA(25⁴, 14, F₂₅, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,25)), using
      - discarding factors / shortening the dual code based on linear OA(25⁴, 25, F₂₅, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
        Reedâ€“Solomon code RS(21,25) [i]

(49−23, 49, 257570)-Net in Base 25 — Upper bound on s

There is no (26, 49, 257571)-net in base 25, because

1 times m-reduction [i] would yield (26, 48, 257571)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 12 622221 042492 419585 418090 309087 854520 030586 198164 436013 847451 904665 > 25⁴⁸ [i]