MinT - Best Known (78−21, 78, s)-Nets in Base 25

Best Known (78−21, 78, s)-Nets in Base 25

(78−21, 78, 1630)-Net over F₂₅ — Constructive and digital

Digital (57, 78, 1630)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 14, 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 (43, 64, 1564)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁶⁴, 1564, F₂₅, 21, 21) (dual of [(1564, 21), 32780, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(25⁶⁴, 15641, F₂₅, 21) (dual of [15641, 15577, 22]-code), using
      - constructionÂ X applied to C([0,10]) âŠ‚ C([0,8]) [i] based on
        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(25⁴⁹, 15626, F₂₅, 17) (dual of [15626, 15577, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626Â | 25⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
        linear OA(25³, 15, F₂₅, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,25) or 15-cap in PG(2,25)), using
        discarding factors / shortening the dual code based on linear OA(25³, 25, F₂₅, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
        Reedâ€“Solomon code RS(22,25) [i]

(78−21, 78, 97973)-Net over F₂₅ — Digital

Digital (57, 78, 97973)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(78−21, 78, large)-Net in Base 25 — Upper bound on s

There is no (57, 78, large)-net in base 25, because

19 times m-reduction [i] would yield (57, 59, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (78−21, 78, s)-Nets in Base 25

(78−21, 78, 1630)-Net over F25 — Constructive and digital

(78−21, 78, 97973)-Net over F25 — Digital

(78−21, 78, large)-Net in Base 25 — Upper bound on s

(78−21, 78, 1630)-Net over F₂₅ — Constructive and digital

(78−21, 78, 97973)-Net over F₂₅ — Digital