MinT - Best Known (83, 83+23, s)-Nets in Base 5

Best Known (83, 83+23, s)-Nets in Base 5

(83, 83+23, 504)-Net over F₅ — Constructive and digital

Digital (83, 106, 504)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (29, 40, 312)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁴⁰, 312, F₅, 11, 11) (dual of [(312, 11), 3392, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(5⁴⁰, 1561, F₅, 11) (dual of [1561, 1521, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(5⁴⁰, 1562, F₅, 11) (dual of [1562, 1522, 12]-code), using
        the BCH-code C(I) with length 1562Â | 5⁵âˆ’1, defining interval IÂ =Â {625,802,979,…,−729}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
2. digital (43, 66, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 33, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
        the Hermitian function field over F₂₅ [i]

(83, 83+23, 5290)-Net over F₅ — Digital

Digital (83, 106, 5290)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(83, 83+23, 5766867)-Net in Base 5 — Upper bound on s

There is no (83, 106, 5766868)-net in base 5, because

1 times m-reduction [i] would yield (83, 105, 5766868)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 24 651918 619318 725961 074918 800584 776246 937479 049694 946273 835874 954700 882769 > 5¹⁰⁵ [i]