MinT - Best Known (58, 58+21, s)-Nets in Base 25

Best Known (58, 58+21, s)-Nets in Base 25

(58, 58+21, 1640)-Net over F₂₅ — Constructive and digital

Digital (58, 79, 1640)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (8, 18, 78)-net over F₂₅, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 5, 26)-net over F₂₅, using
      - net from sequence [i] based on digital (0, 25)-sequence over F₂₅, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 0 and N(F) ≥ 26, using
        the rational function field F₂₅(x) [i]
        Niederreiter sequence [i]
    2. digital (3, 13, 52)-net over F₂₅, using
      - net from sequence [i] based on digital (3, 51)-sequence over F₂₅, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 52, using
        a function field by GarcÃa and Quoos [i]
2. digital (40, 61, 1562)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁶¹, 1562, F₂₅, 21, 21) (dual of [(1562, 21), 32741, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(25⁶¹, 15621, F₂₅, 21) (dual of [15621, 15560, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁶¹, 15625, F₂₅, 21) (dual of [15625, 15564, 22]-code), using
        an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(58, 58+21, 115079)-Net over F₂₅ — Digital

Digital (58, 79, 115079)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(58, 58+21, large)-Net in Base 25 — Upper bound on s

There is no (58, 79, large)-net in base 25, because

19 times m-reduction [i] would yield (58, 60, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (58, 58+21, s)-Nets in Base 25

(58, 58+21, 1640)-Net over F25 — Constructive and digital

(58, 58+21, 115079)-Net over F25 — Digital

(58, 58+21, large)-Net in Base 25 — Upper bound on s

(58, 58+21, 1640)-Net over F₂₅ — Constructive and digital

(58, 58+21, 115079)-Net over F₂₅ — Digital