MinT - Best Known (67−16, 67, s)-Nets in Base 25

Best Known (67−16, 67, s)-Nets in Base 25

(67−16, 67, 48831)-Net over F₂₅ — Constructive and digital

Digital (51, 67, 48831)-net over F₂₅, using

25² times duplication [i] based on digital (49, 65, 48831)-net over F₂₅, using
- net defined by OOA [i] based on linear OOA(25⁶⁵, 48831, F₂₅, 16, 16) (dual of [(48831, 16), 781231, 17]-NRT-code), using
  - OA 8-folding and stacking [i] based on linear OA(25⁶⁵, 390648, F₂₅, 16) (dual of [390648, 390583, 17]-code), using
    - discarding factors / shortening the dual code based on linear OA(25⁶⁵, 390649, F₂₅, 16) (dual of [390649, 390584, 17]-code), using
      - constructionÂ X applied to C_e(15) âŠ‚ C_e(10) [i] based on
        linear OA(25⁶¹, 390625, F₂₅, 16) (dual of [390625, 390564, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 25⁴âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
        linear OA(25⁴¹, 390625, F₂₅, 11) (dual of [390625, 390584, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 25⁴âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [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]

(67−16, 67, 469564)-Net over F₂₅ — Digital

Digital (51, 67, 469564)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(67−16, 67, large)-Net in Base 25 — Upper bound on s

There is no (51, 67, large)-net in base 25, because

14 times m-reduction [i] would yield (51, 53, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]