MinT - Best Known (108−15, 108, s)-Nets in Base 3

Best Known (108−15, 108, s)-Nets in Base 3

(108−15, 108, 8442)-Net over F₃ — Constructive and digital

Digital (93, 108, 8442)-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 (85, 100, 8435)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁰⁰, 8435, F₃, 15, 15) (dual of [(8435, 15), 126425, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(3¹⁰⁰, 59046, F₃, 15) (dual of [59046, 58946, 16]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁰⁰, 59048, F₃, 15) (dual of [59048, 58948, 16]-code), using
        1 times truncation [i] based on linear OA(3¹⁰¹, 59049, F₃, 16) (dual of [59049, 58948, 17]-code), using
        an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(108−15, 108, 29543)-Net over F₃ — Digital

Digital (93, 108, 29543)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁰⁸, 29543, F₃, 2, 15) (dual of [(29543, 2), 58978, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3¹⁰⁸, 59086, F₃, 15) (dual of [59086, 58978, 16]-code), using
  - 1 times truncation [i] based on linear OA(3¹⁰⁹, 59087, F₃, 16) (dual of [59087, 58978, 17]-code), using
    - constructionÂ X applied to C_e(15) âŠ‚ C_e(10) [i] based on
      1. linear OA(3¹⁰¹, 59049, F₃, 16) (dual of [59049, 58948, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
      2. linear OA(3⁷¹, 59049, F₃, 11) (dual of [59049, 58978, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
      3. linear OA(3⁸, 38, F₃, 4) (dual of [38, 30, 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]

(108−15, 108, large)-Net in Base 3 — Upper bound on s

There is no (93, 108, large)-net in base 3, because

13 times m-reduction [i] would yield (93, 95, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]