MinT - Best Known (12, 12+5, s)-Nets in Base 5

Best Known (12, 12+5, s)-Nets in Base 5

(12, 12+5, 631)-Net over F₅ — Constructive and digital

Digital (12, 17, 631)-net over F₅, using

net defined by OOA [i] based on linear OOA(5¹⁷, 631, F₅, 6, 5) (dual of [(631, 6), 3769, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(5¹⁷, 632, F₅, 2, 5) (dual of [(632, 2), 1247, 6]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(5³, 31, F₅, 2, 2) (dual of [(31, 2), 59, 3]-NRT-code), using
      - appending kth column [i] based on linear OA(5³, 31, F₅, 2) (dual of [31, 28, 3]-code), using
        Hamming code H(3,5) [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]

(12, 12+5, 1233)-Net over F₅ — Digital

Digital (12, 17, 1233)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁷, 1233, F₅, 5) (dual of [1233, 1216, 6]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(5³, 31, F₅, 2) (dual of [31, 28, 3]-code), using
    - Hamming code H(3,5) [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]

(12, 12+5, 138105)-Net in Base 5 — Upper bound on s

There is no (12, 17, 138106)-net in base 5, because

1 times m-reduction [i] would yield (12, 16, 138106)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 152588 347585 > 5¹⁶ [i]