MinT - Best Known (70−12, 70, s)-Nets in Base 5

Best Known (70−12, 70, s)-Nets in Base 5

(70−12, 70, 13028)-Net over F₅ — Constructive and digital

Digital (58, 70, 13028)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (0, 6, 6)-net over F₅, using
  - net from sequence [i] based on digital (0, 5)-sequence over F₅, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 0 and N(F) ≥ 6, using
      - the rational function field F₅(x) [i]
    - Niederreiter sequence [i]
2. digital (52, 64, 13022)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁶⁴, 13022, F₅, 12, 12) (dual of [(13022, 12), 156200, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(5⁶⁴, 78132, F₅, 12) (dual of [78132, 78068, 13]-code), using
      - constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
        linear OA(5⁶⁴, 78125, F₅, 12) (dual of [78125, 78061, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 78124Â = 5⁷âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(5⁵⁷, 78125, F₅, 11) (dual of [78125, 78068, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 78124Â = 5⁷âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(5⁰, 7, F₅, 0) (dual of [7, 7, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(5⁰, s, F₅, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(70−12, 70, 75296)-Net over F₅ — Digital

Digital (58, 70, 75296)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁷⁰, 75296, F₅, 12) (dual of [75296, 75226, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁷⁰, 78132, F₅, 12) (dual of [78132, 78062, 13]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(5⁶, 7, F₅, 6) (dual of [7, 1, 7]-code or 7-arc in PG(5,5)), using
      - dual of repetition code with length 7 [i]
    2. linear OA(5⁶⁴, 78125, F₅, 12) (dual of [78125, 78061, 13]-code), using
      - an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 78124Â = 5⁷âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(70−12, 70, large)-Net in Base 5 — Upper bound on s

There is no (58, 70, large)-net in base 5, because

10 times m-reduction [i] would yield (58, 60, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (70−12, 70, s)-Nets in Base 5

(70−12, 70, 13028)-Net over F5 — Constructive and digital

(70−12, 70, 75296)-Net over F5 — Digital

(70−12, 70, large)-Net in Base 5 — Upper bound on s

(70−12, 70, 13028)-Net over F₅ — Constructive and digital

(70−12, 70, 75296)-Net over F₅ — Digital