MinT - Best Known (21−5, 21, s)-Nets in Base 5

Best Known (21−5, 21, s)-Nets in Base 5

(21−5, 21, 2403)-Net over F₅ — Constructive and digital

Digital (16, 21, 2403)-net over F₅, using

net defined by OOA [i] based on linear OOA(5²¹, 2403, F₅, 6, 5) (dual of [(2403, 6), 14397, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(5²¹, 2404, F₅, 2, 5) (dual of [(2404, 2), 4787, 6]-NRT-code), using
  - generalized (u,Â u+v)-construction [i] based on
    1. linear OOA(5¹, 601, F₅, 2, 1) (dual of [(601, 2), 1201, 2]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(5¹, s, F₅, 2, 1) with arbitrarily large s, using
        appending 1 arbitrary column [i] based on linear OA(5¹, s, F₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
    2. linear OOA(5¹, 601, F₅, 2, 1) (dual of [(601, 2), 1201, 2]-NRT-code) (see above)
    3. linear OOA(5⁵, 601, F₅, 2, 2) (dual of [(601, 2), 1197, 3]-NRT-code), using
      - discarding factors / shortening the dual code based on 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]
    4. 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]

(21−5, 21, 3606)-Net over F₅ — Digital

Digital (16, 21, 3606)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5²¹, 3606, F₅, 5) (dual of [3606, 3585, 6]-code), using
- (u,Â uâˆ’v,Â u+v+w)-construction [i] based on
  1. linear OA(5¹, 1202, F₅, 1) (dual of [1202, 1201, 2]-code), using
    - discarding factors / shortening the dual code based on linear OA(5¹, s, F₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
      - parity-check code [i]
  2. linear OA(5⁶, 1202, F₅, 2) (dual of [1202, 1196, 3]-code), using
    - discarding factors / shortening the dual code based on linear OA(5⁶, 3125, F₅, 2) (dual of [3125, 3119, 3]-code), using
      - an extension C_e(1) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 2 [i]
  3. 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]

(21−5, 21, 3452668)-Net in Base 5 — Upper bound on s

There is no (16, 21, 3452669)-net in base 5, because

1 times m-reduction [i] would yield (16, 20, 3452669)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 95 367441 031193 > 5²⁰ [i]