MinT - Best Known (25, 25+5, s)-Nets in Base 3

Best Known (25, 25+5, s)-Nets in Base 3

(25, 25+5, 12965)-Net over F₃ — Constructive and digital

Digital (25, 30, 12965)-net over F₃, using

net defined by OOA [i] based on linear OOA(3³⁰, 12965, F₃, 5, 5) (dual of [(12965, 5), 64795, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(3³⁰, 12966, F₃, 2, 5) (dual of [(12966, 2), 25902, 6]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(3², 4, F₃, 2, 2) (dual of [(4, 2), 6, 3]-NRT-code), using
      - extended Reedâ€“Solomon NRT-code RS_e(2;6,3) [i]
    2. linear OOA(3²⁸, 12962, F₃, 2, 5) (dual of [(12962, 2), 25896, 6]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(3²⁸, 25924, F₃, 5) (dual of [25924, 25896, 6]-code), using
        trace code [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]

(25, 25+5, 25928)-Net over F₃ — Digital

Digital (25, 30, 25928)-net over F₃, using

net defined by OOA [i] based on linear OOA(3³⁰, 25928, F₃, 5, 5) (dual of [(25928, 5), 129610, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(3³⁰, 25928, F₃, 4, 5) (dual of [(25928, 4), 103682, 6]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3³⁰, 25928, F₃, 5) (dual of [25928, 25898, 6]-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OA(3², 4, F₃, 2) (dual of [4, 2, 3]-code or 4-arc in PG(1,3)), using
        extended Reedâ€“Solomon code RS_e(2,3) [i]
        Hamming code H(2,3) [i]
        Simplex code S(2,3) [i]
        the Tetracode [i]
      2. linear OA(3²⁸, 25924, F₃, 5) (dual of [25924, 25896, 6]-code), using
        trace code [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]

(25, 25+5, 5857915)-Net in Base 3 — Upper bound on s

There is no (25, 30, 5857916)-net in base 3, because

1 times m-reduction [i] would yield (25, 29, 5857916)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 68 630394 873609 > 3²⁹ [i]