MinT - Best Known (56−11, 56, s)-Nets in Base 25

Best Known (56−11, 56, s)-Nets in Base 25

(56−11, 56, 1677746)-Net over F₂₅ — Constructive and digital

Digital (45, 56, 1677746)-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 (40, 51, 1677720)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁵¹, 1677720, F₂₅, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(25⁵¹, 8388601, F₂₅, 11) (dual of [8388601, 8388550, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁵¹, large, F₂₅, 11) (dual of [large, large−51, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(56−11, 56, large)-Net over F₂₅ — Digital

Digital (45, 56, large)-net over F₂₅, using

t-expansion [i] based on digital (44, 56, large)-net over F₂₅, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] 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]

(56−11, 56, large)-Net in Base 25 — Upper bound on s

There is no (45, 56, large)-net in base 25, because

9 times m-reduction [i] would yield (45, 47, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]