MinT - Best Known (68, 79, s)-Nets in Base 5

Best Known (68, 79, s)-Nets in Base 5

(68, 79, 390635)-Net over F₅ — Constructive and digital

Digital (68, 79, 390635)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (1, 6, 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 (62, 73, 390625)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁷³, 390625, F₅, 11, 11) (dual of [(390625, 11), 4296802, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(5⁷³, 1953126, F₅, 11) (dual of [1953126, 1953053, 12]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 1953126Â | 5¹⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(68, 79, 1184212)-Net over F₅ — Digital

Digital (68, 79, 1184212)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁷⁹, 1184212, F₅, 11) (dual of [1184212, 1184133, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁷⁹, 1953138, F₅, 11) (dual of [1953138, 1953059, 12]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(5⁶, 12, F₅, 5) (dual of [12, 6, 6]-code), using
      - extended quadratic residue code Q_e(12,5) [i]
    2. linear OA(5⁷³, 1953126, F₅, 11) (dual of [1953126, 1953053, 12]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 1953126Â | 5¹⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(68, 79, large)-Net in Base 5 — Upper bound on s

There is no (68, 79, large)-net in base 5, because

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