MinT - Best Known (109−16, 109, s)-Nets in Base 25

Best Known (109−16, 109, s)-Nets in Base 25

(109−16, 109, 1146258)-Net over F₂₅ — Constructive and digital

Digital (93, 109, 1146258)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (25, 33, 97683)-net over F₂₅, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 4, 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 (21, 29, 97657)-net over F₂₅, using
      - net defined by OOA [i] based on linear OOA(25²⁹, 97657, F₂₅, 8, 8) (dual of [(97657, 8), 781227, 9]-NRT-code), using
        OA 4-folding and stacking [i] based on linear OA(25²⁹, 390628, F₂₅, 8) (dual of [390628, 390599, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(25²⁹, 390629, F₂₅, 8) (dual of [390629, 390600, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(25²⁹, 390625, F₂₅, 8) (dual of [390625, 390596, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 25⁴âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(25²⁵, 390625, F₂₅, 7) (dual of [390625, 390600, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 25⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(25⁰, 4, F₂₅, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁰, s, F₂₅, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. digital (60, 76, 1048575)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁷⁶, 1048575, F₂₅, 16, 16) (dual of [(1048575, 16), 16777124, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(25⁷⁶, 8388600, F₂₅, 16) (dual of [8388600, 8388524, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁷⁶, large, F₂₅, 16) (dual of [large, large−76, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(109−16, 109, large)-Net over F₂₅ — Digital

Digital (93, 109, large)-net over F₂₅, using

25² times duplication [i] based on digital (91, 107, large)-net over F₂₅, using
- t-expansion [i] based on digital (85, 107, large)-net over F₂₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25¹⁰⁷, large, F₂₅, 22) (dual of [large, large−107, 23]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(25¹⁰⁶, large, F₂₅, 22) (dual of [large, large−106, 23]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(109−16, 109, large)-Net in Base 25 — Upper bound on s

There is no (93, 109, large)-net in base 25, because

14 times m-reduction [i] would yield (93, 95, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (109−16, 109, s)-Nets in Base 25

(109−16, 109, 1146258)-Net over F25 — Constructive and digital

(109−16, 109, large)-Net over F25 — Digital

(109−16, 109, large)-Net in Base 25 — Upper bound on s

(109−16, 109, 1146258)-Net over F₂₅ — Constructive and digital

(109−16, 109, large)-Net over F₂₅ — Digital