MinT - Best Known (50−12, 50, s)-Nets in Base 25

Best Known (50−12, 50, s)-Nets in Base 25

(50−12, 50, 65108)-Net over F₂₅ — Constructive and digital

Digital (38, 50, 65108)-net over F₂₅, using

25¹ times duplication [i] based on digital (37, 49, 65108)-net over F₂₅, using
- net defined by OOA [i] based on linear OOA(25⁴⁹, 65108, F₂₅, 12, 12) (dual of [(65108, 12), 781247, 13]-NRT-code), using
  - OA 6-folding and stacking [i] based on linear OA(25⁴⁹, 390648, F₂₅, 12) (dual of [390648, 390599, 13]-code), using
    - discarding factors / shortening the dual code based on linear OA(25⁴⁹, 390649, F₂₅, 12) (dual of [390649, 390600, 13]-code), using
      - constructionÂ X applied to C_e(11) âŠ‚ C_e(6) [i] based on
        linear OA(25⁴⁵, 390625, F₂₅, 12) (dual of [390625, 390580, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 25⁴âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(25²⁵, 390625, F₂₅, 7) (dual of [390625, 390600, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 25⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(25⁴, 24, F₂₅, 4) (dual of [24, 20, 5]-code or 24-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]

(50−12, 50, 462467)-Net over F₂₅ — Digital

Digital (38, 50, 462467)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(50−12, 50, large)-Net in Base 25 — Upper bound on s

There is no (38, 50, large)-net in base 25, because

10 times m-reduction [i] would yield (38, 40, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]