MinT - Best Known (100−16, 100, s)-Nets in Base 3

Best Known (100−16, 100, s)-Nets in Base 3

(100−16, 100, 2467)-Net over F₃ — Constructive and digital

Digital (84, 100, 2467)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 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, 91, 2460)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3⁹¹, 2460, F₃, 16, 16) (dual of [(2460, 16), 39269, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(3⁹¹, 19680, F₃, 16) (dual of [19680, 19589, 17]-code), using
      - discarding factors / shortening the dual code 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]

(100−16, 100, 9859)-Net over F₃ — Digital

Digital (84, 100, 9859)-net over F₃, using

3¹ times duplication [i] based on digital (83, 99, 9859)-net over F₃, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3⁹⁹, 9859, F₃, 2, 16) (dual of [(9859, 2), 19619, 17]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3⁹⁹, 19718, F₃, 16) (dual of [19718, 19619, 17]-code), using
    - constructionÂ X applied to C_e(15) âŠ‚ C_e(10) [i] based on
      1. 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]
      2. linear OA(3⁶⁴, 19683, F₃, 11) (dual of [19683, 19619, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
      3. linear OA(3⁸, 35, F₃, 4) (dual of [35, 27, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁸, 41, F₃, 4) (dual of [41, 33, 5]-code), using
        the narrow-sense BCH-code C(I) with length 41Â | 3⁸âˆ’1, defining interval IÂ =Â [1,1], and minimum distance dÂ â‰¥ |{−3,−1,1,3}|+1Â =Â 5 (BCH-bound) [i]

(100−16, 100, 1732502)-Net in Base 3 — Upper bound on s

There is no (84, 100, 1732503)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 515378 675143 647887 026649 097604 804800 005169 326193 > 3¹⁰⁰ [i]