MinT - Best Known (52, 52+11, s)-Nets in Base 5

Best Known (52, 52+11, s)-Nets in Base 5

(52, 52+11, 15635)-Net over F₅ — Constructive and digital

Digital (52, 63, 15635)-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 (46, 57, 15625)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁵⁷, 15625, F₅, 11, 11) (dual of [(15625, 11), 171818, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(5⁵⁷, 78126, F₅, 11) (dual of [78126, 78069, 12]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 78126Â | 5¹⁴âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(52, 52+11, 67730)-Net over F₅ — Digital

Digital (52, 63, 67730)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁶³, 67730, F₅, 11) (dual of [67730, 67667, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁶³, 78138, F₅, 11) (dual of [78138, 78075, 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(5⁵⁷, 78126, F₅, 11) (dual of [78126, 78069, 12]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 78126Â | 5¹⁴âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(52, 52+11, large)-Net in Base 5 — Upper bound on s

There is no (52, 63, large)-net in base 5, because

9 times m-reduction [i] would yield (52, 54, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]