MinT - Best Known (223−45, 223, s)-Nets in Base 3

Best Known (223−45, 223, s)-Nets in Base 3

(223−45, 223, 688)-Net over F₃ — Constructive and digital

Digital (178, 223, 688)-net over F₃, using

5 times m-reduction [i] based on digital (178, 228, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 57, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(223−45, 223, 2322)-Net over F₃ — Digital

Digital (178, 223, 2322)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²³, 2322, F₃, 45) (dual of [2322, 2099, 46]-code), using
- 108 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 12 times 0, 1, 17 times 0, 1, 24 times 0, 1, 31 times 0) [i] based on linear OA(3²¹², 2203, F₃, 45) (dual of [2203, 1991, 46]-code), using
  - constructionÂ X applied to C([0,22]) âŠ‚ C([0,21]) [i] based on
    1. linear OA(3²¹¹, 2188, F₃, 45) (dual of [2188, 1977, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,22], and minimum distance dÂ â‰¥ |{−22,−21,…,22}|+1Â =Â 46 (BCH-bound) [i]
    2. linear OA(3¹⁹⁷, 2188, F₃, 43) (dual of [2188, 1991, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,21], and minimum distance dÂ â‰¥ |{−21,−20,…,21}|+1Â =Â 44 (BCH-bound) [i]
    3. linear OA(3¹, 15, F₃, 1) (dual of [15, 14, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(223−45, 223, 295378)-Net in Base 3 — Upper bound on s

There is no (178, 223, 295379)-net in base 3, because

1 times m-reduction [i] would yield (178, 222, 295379)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8335 665230 514361 888394 308913 205479 535949 541617 683548 482249 635788 659263 220692 866253 705170 694038 304376 524317 > 3²²² [i]