MinT - Best Known (13−4, 13, s)-Nets in Base 5

Best Known (13−4, 13, s)-Nets in Base 5

(13−4, 13, 600)-Net over F₅ — Constructive and digital

Digital (9, 13, 600)-net over F₅, using

1 times m-reduction [i] based on digital (9, 14, 600)-net over F₅, using
- net defined by OOA [i] based on linear OOA(5¹⁴, 600, F₅, 5, 5) (dual of [(600, 5), 2986, 6]-NRT-code), using
  - OOA 2-folding and stacking with additional row [i] based on linear OA(5¹⁴, 1201, F₅, 5) (dual of [1201, 1187, 6]-code), using
    - discarding factors / shortening the dual code 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]

(13−4, 13, 1221)-Net over F₅ — Digital

Digital (9, 13, 1221)-net over F₅, using

net defined by OOA [i] based on linear OOA(5¹³, 1221, F₅, 4, 4) (dual of [(1221, 4), 4871, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(5¹³, 1221, F₅, 3, 4) (dual of [(1221, 3), 3650, 5]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹³, 1221, F₅, 4) (dual of [1221, 1208, 5]-code), using
    - 20 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 19 times 0) [i] based on linear OA(5¹², 1200, F₅, 4) (dual of [1200, 1188, 5]-code), using
      - trace code [i] based on linear OA(25⁶, 600, F₂₅, 4) (dual of [600, 594, 5]-code), using
        1 times truncation [i] based on linear OA(25⁷, 601, F₂₅, 5) (dual of [601, 594, 6]-code), using
        a code by Danev and Olsson [i]

(13−4, 13, 12351)-Net in Base 5 — Upper bound on s

There is no (9, 13, 12352)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1220 772865 > 5¹³ [i]