MinT - Best Known (150−13, 150, s)-Nets in Base 5

Best Known (150−13, 150, s)-Nets in Base 5

(150−13, 150, 2801414)-Net over F₅ — Constructive and digital

Digital (137, 150, 2801414)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (22, 28, 5214)-net over F₅, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 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 (19, 25, 5208)-net over F₅, using
      - net defined by OOA [i] based on linear OOA(5²⁵, 5208, F₅, 6, 6) (dual of [(5208, 6), 31223, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(5²⁵, 15624, F₅, 6) (dual of [15624, 15599, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(5²⁵, 15625, F₅, 6) (dual of [15625, 15600, 7]-code), using
        an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 5⁶âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
2. digital (109, 122, 2796200)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5¹²², 2796200, F₅, 14, 13) (dual of [(2796200, 14), 39146678, 14]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(5¹²², 8388601, F₅, 2, 13) (dual of [(8388601, 2), 16777080, 14]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(5¹²², 8388602, F₅, 2, 13) (dual of [(8388602, 2), 16777082, 14]-NRT-code), using
        trace code [i] based on linear OOA(25⁶¹, 4194301, F₂₅, 2, 13) (dual of [(4194301, 2), 8388541, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(25⁶¹, 8388602, F₂₅, 13) (dual of [8388602, 8388541, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁶¹, large, F₂₅, 13) (dual of [large, large−61, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(150−13, 150, large)-Net over F₅ — Digital

Digital (137, 150, large)-net over F₅, using

5⁴ times duplication [i] based on digital (133, 146, large)-net over F₅, using
- t-expansion [i] based on digital (129, 146, large)-net over F₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁴⁶, large, F₅, 17) (dual of [large, large−146, 18]-code), using
    - 15 times code embedding in larger space [i] based on linear OA(5¹³¹, large, F₅, 17) (dual of [large, large−131, 18]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(150−13, 150, large)-Net in Base 5 — Upper bound on s

There is no (137, 150, large)-net in base 5, because

11 times m-reduction [i] would yield (137, 139, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (150−13, 150, s)-Nets in Base 5

(150−13, 150, 2801414)-Net over F5 — Constructive and digital

(150−13, 150, large)-Net over F5 — Digital

(150−13, 150, large)-Net in Base 5 — Upper bound on s

(150−13, 150, 2801414)-Net over F₅ — Constructive and digital

(150−13, 150, large)-Net over F₅ — Digital