MinT - Best Known (80−12, 80, s)-Nets in Base 25

Best Known (80−12, 80, s)-Nets in Base 25

(80−12, 80, 1528335)-Net over F₂₅ — Constructive and digital

Digital (68, 80, 1528335)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (18, 24, 130235)-net over F₂₅, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 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 (15, 21, 130209)-net over F₂₅, using
      - net defined by OOA [i] based on linear OOA(25²¹, 130209, F₂₅, 6, 6) (dual of [(130209, 6), 781233, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(25²¹, 390627, F₂₅, 6) (dual of [390627, 390606, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(25²¹, 390629, F₂₅, 6) (dual of [390629, 390608, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(25²¹, 390625, F₂₅, 6) (dual of [390625, 390604, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 25⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(25¹⁷, 390625, F₂₅, 5) (dual of [390625, 390608, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 25⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [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 (44, 56, 1398100)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁵⁶, 1398100, F₂₅, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(25⁵⁶, 8388600, F₂₅, 12) (dual of [8388600, 8388544, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁵⁶, large, F₂₅, 12) (dual of [large, large−56, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(80−12, 80, large)-Net over F₂₅ — Digital

Digital (68, 80, large)-net over F₂₅, using

t-expansion [i] based on digital (65, 80, large)-net over F₂₅, using
- 2 times m-reduction [i] based on digital (65, 82, large)-net over F₂₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁸², large, F₂₅, 17) (dual of [large, large−82, 18]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(25⁸¹, large, F₂₅, 17) (dual of [large, large−81, 18]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(80−12, 80, large)-Net in Base 25 — Upper bound on s

There is no (68, 80, large)-net in base 25, because

10 times m-reduction [i] would yield (68, 70, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (80−12, 80, s)-Nets in Base 25

(80−12, 80, 1528335)-Net over F25 — Constructive and digital

(80−12, 80, large)-Net over F25 — Digital

(80−12, 80, large)-Net in Base 25 — Upper bound on s

(80−12, 80, 1528335)-Net over F₂₅ — Constructive and digital

(80−12, 80, large)-Net over F₂₅ — Digital