MinT - Best Known (72, 72+29, s)-Nets in Base 25

Best Known (72, 72+29, s)-Nets in Base 25

(72, 72+29, 1182)-Net over F₂₅ — Constructive and digital

Digital (72, 101, 1182)-net over F₂₅, using

25¹ times duplication [i] based on digital (71, 100, 1182)-net over F₂₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (4, 18, 66)-net over F₂₅, using
    - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
        a function field by GarcÃa and Quoos [i]
  2. digital (53, 82, 1116)-net over F₂₅, using
    - net defined by OOA [i] based on linear OOA(25⁸², 1116, F₂₅, 29, 29) (dual of [(1116, 29), 32282, 30]-NRT-code), using
      - OOA 14-folding and stacking with additional row [i] based on linear OA(25⁸², 15625, F₂₅, 29) (dual of [15625, 15543, 30]-code), using
        an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(72, 72+29, 51937)-Net over F₂₅ — Digital

Digital (72, 101, 51937)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+29, large)-Net in Base 25 — Upper bound on s

There is no (72, 101, large)-net in base 25, because

27 times m-reduction [i] would yield (72, 74, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]