MinT - Best Known (70, 70+23, s)-Nets in Base 5

Best Known (70, 70+23, s)-Nets in Base 5

(70, 70+23, 312)-Net over F₅ — Constructive and digital

Digital (70, 93, 312)-net over F₅, using

5¹ times duplication [i] based on digital (69, 92, 312)-net over F₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (17, 28, 104)-net over F₅, using
    - trace code for nets [i] based on digital (3, 14, 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 (41, 64, 208)-net over F₅, using
    - trace code for nets [i] based on digital (9, 32, 104)-net over F₂₅, using
      - net from sequence [i] based on digital (9, 103)-sequence over F₂₅, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 9 and N(F) ≥ 104, using
        a function field by GarcÃa and Quoos [i]

(70, 70+23, 2489)-Net over F₅ — Digital

Digital (70, 93, 2489)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁹³, 2489, F₅, 23) (dual of [2489, 2396, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁹³, 3138, F₅, 23) (dual of [3138, 3045, 24]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(5⁹², 3137, F₅, 23) (dual of [3137, 3045, 24]-code), using
    - constructionÂ X applied to C([0,11]) âŠ‚ C([0,10]) [i] based on
      1. linear OA(5⁹¹, 3126, F₅, 23) (dual of [3126, 3035, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126Â | 5¹⁰âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
      2. linear OA(5⁸¹, 3126, F₅, 21) (dual of [3126, 3045, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126Â | 5¹⁰âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
      3. linear OA(5¹, 11, F₅, 1) (dual of [11, 10, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(5¹, s, F₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(70, 70+23, 860758)-Net in Base 5 — Upper bound on s

There is no (70, 93, 860759)-net in base 5, because

1 times m-reduction [i] would yield (70, 92, 860759)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 20195 029628 287669 084158 181177 978248 729726 126267 547180 095118 265525 > 5⁹² [i]