MinT - Best Known (80−12, 80, s)-Nets in Base 5

Best Known (80−12, 80, s)-Nets in Base 5

(80−12, 80, 65115)-Net over F₅ — Constructive and digital

Digital (68, 80, 65115)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (1, 7, 10)-net over F₅, using
  - net from sequence [i] based on digital (1, 9)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 1 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (61, 73, 65105)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁷³, 65105, F₅, 12, 12) (dual of [(65105, 12), 781187, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(5⁷³, 390630, F₅, 12) (dual of [390630, 390557, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(5⁷³, 390633, F₅, 12) (dual of [390633, 390560, 13]-code), using
        constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
        linear OA(5⁷³, 390625, F₅, 12) (dual of [390625, 390552, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 5⁸âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(5⁶⁵, 390625, F₅, 11) (dual of [390625, 390560, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 5⁸âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(5⁰, 8, F₅, 0) (dual of [8, 8, 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]

(80−12, 80, 376506)-Net over F₅ — Digital

Digital (68, 80, 376506)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁸⁰, 376506, F₅, 12) (dual of [376506, 376426, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁸⁰, 390635, F₅, 12) (dual of [390635, 390555, 13]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(5⁷, 10, F₅, 6) (dual of [10, 3, 7]-code), using
      - discarding factors / shortening the dual code based on linear OA(5⁷, 11, F₅, 6) (dual of [11, 4, 7]-code), using
        1 times truncation [i] based on linear OA(5⁸, 12, F₅, 7) (dual of [12, 4, 8]-code), using
        an explicit generator matrix [i]
    2. linear OA(5⁷³, 390625, F₅, 12) (dual of [390625, 390552, 13]-code), using
      - an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 5⁸âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(80−12, 80, large)-Net in Base 5 — Upper bound on s

There is no (68, 80, large)-net in base 5, because

10 times m-reduction [i] would yield (68, 70, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (80−12, 80, s)-Nets in Base 5

(80−12, 80, 65115)-Net over F5 — Constructive and digital

(80−12, 80, 376506)-Net over F5 — Digital

(80−12, 80, large)-Net in Base 5 — Upper bound on s

(80−12, 80, 65115)-Net over F₅ — Constructive and digital

(80−12, 80, 376506)-Net over F₅ — Digital