MinT - Best Known (82−22, 82, s)-Nets in Base 25

Best Known (82−22, 82, s)-Nets in Base 25

(82−22, 82, 1487)-Net over F₂₅ — Constructive and digital

Digital (60, 82, 1487)-net over F₂₅, using

1 times m-reduction [i] based on digital (60, 83, 1487)-net over F₂₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (4, 15, 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 (45, 68, 1421)-net over F₂₅, using
    - net defined by OOA [i] based on linear OOA(25⁶⁸, 1421, F₂₅, 23, 23) (dual of [(1421, 23), 32615, 24]-NRT-code), using
      - OOA 11-folding and stacking with additional row [i] based on linear OA(25⁶⁸, 15632, F₂₅, 23) (dual of [15632, 15564, 24]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁶⁸, 15633, F₂₅, 23) (dual of [15633, 15565, 24]-code), using
        constructionÂ X applied to C([0,11]) âŠ‚ C([0,10]) [i] based on
        linear OA(25⁶⁷, 15626, F₂₅, 23) (dual of [15626, 15559, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626Â | 25⁶âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
        linear OA(25⁶¹, 15626, F₂₅, 21) (dual of [15626, 15565, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626Â | 25⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(25¹, 7, F₂₅, 1) (dual of [7, 6, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(25¹, s, F₂₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(82−22, 82, 103980)-Net over F₂₅ — Digital

Digital (60, 82, 103980)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(82−22, 82, large)-Net in Base 25 — Upper bound on s

There is no (60, 82, large)-net in base 25, because

20 times m-reduction [i] would yield (60, 62, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (82−22, 82, s)-Nets in Base 25

(82−22, 82, 1487)-Net over F25 — Constructive and digital

(82−22, 82, 103980)-Net over F25 — Digital

(82−22, 82, large)-Net in Base 25 — Upper bound on s

(82−22, 82, 1487)-Net over F₂₅ — Constructive and digital

(82−22, 82, 103980)-Net over F₂₅ — Digital