MinT - Best Known (16−5, 16, s)-Nets in Base 3

Best Known (16−5, 16, s)-Nets in Base 3

(16−5, 16, 157)-Net over F₃ — Constructive and digital

Digital (11, 16, 157)-net over F₃, using

net defined by OOA [i] based on linear OOA(3¹⁶, 157, F₃, 5, 5) (dual of [(157, 5), 769, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(3¹⁶, 157, F₃, 4, 5) (dual of [(157, 4), 612, 6]-NRT-code), using
  - generalized (u,Â u+v)-construction [i] based on
    1. linear OOA(3¹, 40, F₃, 4, 1) (dual of [(40, 4), 159, 2]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(3¹, s, F₃, 4, 1) with arbitrarily large s, using
        appending 3 arbitrary columns [i] based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
    2. linear OOA(3⁴, 40, F₃, 4, 2) (dual of [(40, 4), 156, 3]-NRT-code), using
      - appending 2 arbitrary columns [i] based on linear OOA(3⁴, 40, F₃, 2, 2) (dual of [(40, 2), 76, 3]-NRT-code), using
        appending kth column [i] based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]
    3. linear OOA(3¹¹, 77, F₃, 4, 5) (dual of [(77, 4), 297, 6]-NRT-code), using
      - extracting embedded OOA [i] based on digital (6, 11, 77)-net over F₃, using
        linear code embedding found in a computer search [i]

(16−5, 16, 219)-Net over F₃ — Digital

Digital (11, 16, 219)-net over F₃, using

net defined by OOA [i] based on linear OOA(3¹⁶, 219, F₃, 5, 5) (dual of [(219, 5), 1079, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(3¹⁶, 219, F₃, 4, 5) (dual of [(219, 4), 860, 6]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁶, 219, F₃, 5) (dual of [219, 203, 6]-code), using
    - discarding factors / shortening the dual code based on linear OA(3¹⁶, 242, F₃, 5) (dual of [242, 226, 6]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â [0,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]

(16−5, 16, 2677)-Net in Base 3 — Upper bound on s

There is no (11, 16, 2678)-net in base 3, because

1 times m-reduction [i] would yield (11, 15, 2678)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 14 359437 > 3¹⁵ [i]