MinT - Best Known (131−17, 131, s)-Nets in Base 3

Best Known (131−17, 131, s)-Nets in Base 3

(131−17, 131, 22151)-Net over F₃ — Constructive and digital

Digital (114, 131, 22151)-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 (105, 122, 22144)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹²², 22144, F₃, 17, 17) (dual of [(22144, 17), 376326, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(3¹²², 177153, F₃, 17) (dual of [177153, 177031, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹²², 177158, F₃, 17) (dual of [177158, 177036, 18]-code), using
        constructionÂ X applied to C_e(16) âŠ‚ C_e(15) [i] based on
        linear OA(3¹²², 177147, F₃, 17) (dual of [177147, 177025, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
        linear OA(3¹¹¹, 177147, F₃, 16) (dual of [177147, 177036, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
        linear OA(3⁰, 11, F₃, 0) (dual of [11, 11, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(131−17, 131, 75281)-Net over F₃ — Digital

Digital (114, 131, 75281)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹³¹, 75281, F₃, 2, 17) (dual of [(75281, 2), 150431, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹³¹, 88594, F₃, 2, 17) (dual of [(88594, 2), 177057, 18]-NRT-code), using
  - 3¹ times duplication [i] based on linear OOA(3¹³⁰, 88594, F₃, 2, 17) (dual of [(88594, 2), 177058, 18]-NRT-code), using
    - 1 times NRT-code embedding in larger space [i] based on linear OOA(3¹²⁸, 88593, F₃, 2, 17) (dual of [(88593, 2), 177058, 18]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(3¹²⁸, 177186, F₃, 17) (dual of [177186, 177058, 18]-code), using
        constructionÂ X applied to C_e(16) âŠ‚ C_e(12) [i] based on
        linear OA(3¹²², 177147, F₃, 17) (dual of [177147, 177025, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
        linear OA(3⁸⁹, 177147, F₃, 13) (dual of [177147, 177058, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(3⁶, 39, F₃, 3) (dual of [39, 33, 4]-code or 39-cap in PG(5,3)), using
        discarding factors / shortening the dual code based on linear OA(3⁶, 48, F₃, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
        large caps in PG(5,Â b) [i]

(131−17, 131, large)-Net in Base 3 — Upper bound on s

There is no (114, 131, large)-net in base 3, because

15 times m-reduction [i] would yield (114, 116, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (131−17, 131, s)-Nets in Base 3

(131−17, 131, 22151)-Net over F3 — Constructive and digital

(131−17, 131, 75281)-Net over F3 — Digital

(131−17, 131, large)-Net in Base 3 — Upper bound on s

(131−17, 131, 22151)-Net over F₃ — Constructive and digital

(131−17, 131, 75281)-Net over F₃ — Digital