MinT - Best Known (31, 43, s)-Nets in Base 3

Best Known (31, 43, s)-Nets in Base 3

(31, 43, 144)-Net over F₃ — Constructive and digital

Digital (31, 43, 144)-net over F₃, using

3¹ times duplication [i] based on digital (30, 42, 144)-net over F₃, using
- trace code for nets [i] based on digital (2, 14, 48)-net over F₂₇, using
  - net from sequence [i] based on digital (2, 47)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
      - a function field by GarcÃa and Quoos [i]

(31, 43, 220)-Net over F₃ — Digital

Digital (31, 43, 220)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁴³, 220, F₃, 12) (dual of [220, 177, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁴³, 248, F₃, 12) (dual of [248, 205, 13]-code), using
  - constructionÂ X applied to C([0,6]) âŠ‚ C([0,4]) [i] based on
    1. linear OA(3⁴¹, 244, F₃, 13) (dual of [244, 203, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 244Â | 3¹⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
    2. linear OA(3³¹, 244, F₃, 9) (dual of [244, 213, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 244Â | 3¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
    3. 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]

(31, 43, 3926)-Net in Base 3 — Upper bound on s

There is no (31, 43, 3927)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 328 743916 827059 177445 > 3⁴³ [i]