MinT - Best Known (83, 83+15, s)-Nets in Base 3

Best Known (83, 83+15, s)-Nets in Base 3

(83, 83+15, 2818)-Net over F₃ — Constructive and digital

Digital (83, 98, 2818)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 8, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
2. digital (75, 90, 2811)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3⁹⁰, 2811, F₃, 15, 15) (dual of [(2811, 15), 42075, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(3⁹⁰, 19678, F₃, 15) (dual of [19678, 19588, 16]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁹⁰, 19682, F₃, 15) (dual of [19682, 19592, 16]-code), using
        1 times truncation [i] based on linear OA(3⁹¹, 19683, F₃, 16) (dual of [19683, 19592, 17]-code), using
        an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(83, 83+15, 10278)-Net over F₃ — Digital

Digital (83, 98, 10278)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁹⁸, 10278, F₃, 15) (dual of [10278, 10180, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁹⁸, 19692, F₃, 15) (dual of [19692, 19594, 16]-code), using
  - constructionÂ X applied to C([0,7]) âŠ‚ C([1,7]) [i] based on
    1. linear OA(3⁹¹, 19684, F₃, 15) (dual of [19684, 19593, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
    2. linear OA(3⁹⁰, 19684, F₃, 7) (dual of [19684, 19594, 8]-code), using the narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
    3. linear OA(3⁷, 8, F₃, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,3)), using
      - dual of repetition code with length 8 [i]

(83, 83+15, 6909166)-Net in Base 3 — Upper bound on s

There is no (83, 98, 6909167)-net in base 3, because

1 times m-reduction [i] would yield (83, 97, 6909167)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 19088 074597 215476 371085 284768 106490 580839 639219 > 3⁹⁷ [i]