MinT - Best Known (19−5, 19, s)-Nets in Base 5

Best Known (19−5, 19, s)-Nets in Base 5

(19−5, 19, 1201)-Net over F₅ — Constructive and digital

Digital (14, 19, 1201)-net over F₅, using

net defined by OOA [i] based on linear OOA(5¹⁹, 1201, F₅, 6, 5) (dual of [(1201, 6), 7187, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(5¹⁹, 1202, F₅, 2, 5) (dual of [(1202, 2), 2385, 6]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(5⁵, 781, F₅, 2, 2) (dual of [(781, 2), 1557, 3]-NRT-code), using
      - appending kth column [i] based on linear OA(5⁵, 781, F₅, 2) (dual of [781, 776, 3]-code), using
        Hamming code H(5,5) [i]
    2. linear OOA(5¹⁴, 601, F₅, 2, 5) (dual of [(601, 2), 1188, 6]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(5¹⁴, 1202, F₅, 5) (dual of [1202, 1188, 6]-code), using
        trace code [i] based on linear OA(25⁷, 601, F₂₅, 5) (dual of [601, 594, 6]-code), using
        a code by Danev and Olsson [i]

(19−5, 19, 1983)-Net over F₅ — Digital

Digital (14, 19, 1983)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁹, 1983, F₅, 5) (dual of [1983, 1964, 6]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(5⁵, 781, F₅, 2) (dual of [781, 776, 3]-code), using
    - Hamming code H(5,5) [i]
  2. linear OA(5¹⁴, 1202, F₅, 5) (dual of [1202, 1188, 6]-code), using
    - trace code [i] based on linear OA(25⁷, 601, F₂₅, 5) (dual of [601, 594, 6]-code), using
      - a code by Danev and Olsson [i]

(19−5, 19, 690532)-Net in Base 5 — Upper bound on s

There is no (14, 19, 690533)-net in base 5, because

1 times m-reduction [i] would yield (14, 18, 690533)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3 814697 641241 > 5¹⁸ [i]