MinT - Best Known (224−21, 224, s)-Nets in Base 3

Best Known (224−21, 224, s)-Nets in Base 3

(224−21, 224, 838872)-Net over F₃ — Constructive and digital

Digital (203, 224, 838872)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (4, 14, 12)-net over F₃, using
  - net from sequence [i] based on digital (4, 11)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 4 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
2. digital (189, 210, 838860)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²¹⁰, 838860, F₃, 21, 21) (dual of [(838860, 21), 17615850, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(3²¹⁰, 8388601, F₃, 21) (dual of [8388601, 8388391, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²¹⁰, large, F₃, 21) (dual of [large, large−210, 22]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(224−21, 224, 2894630)-Net over F₃ — Digital

Digital (203, 224, 2894630)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²²⁴, 2894630, F₃, 2, 21) (dual of [(2894630, 2), 5789036, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²²⁴, 4194313, F₃, 2, 21) (dual of [(4194313, 2), 8388402, 22]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(3¹⁴, 12, F₃, 2, 10) (dual of [(12, 2), 10, 11]-NRT-code), using
      - extended algebraic-geometric NRT-code AG_e(2;F,13P) [i] based on function field F/F₃ with g(F) = 4 and N(F) ≥ 12, using
        an explicitly constructive algebraic function field [i]
    2. linear OOA(3²¹⁰, 4194301, F₃, 2, 21) (dual of [(4194301, 2), 8388392, 22]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(3²¹⁰, 8388602, F₃, 21) (dual of [8388602, 8388392, 22]-code), using
        discarding factors / shortening the dual code based on linear OA(3²¹⁰, large, F₃, 21) (dual of [large, large−210, 22]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(224−21, 224, large)-Net in Base 3 — Upper bound on s

There is no (203, 224, large)-net in base 3, because

19 times m-reduction [i] would yield (203, 205, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]