MinT - Best Known (179−21, 179, s)-Nets in Base 3

Best Known (179−21, 179, s)-Nets in Base 3

(179−21, 179, 53150)-Net over F₃ — Constructive and digital

Digital (158, 179, 53150)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 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 (147, 168, 53143)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁶⁸, 53143, F₃, 21, 21) (dual of [(53143, 21), 1115835, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(3¹⁶⁸, 531431, F₃, 21) (dual of [531431, 531263, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁶⁸, 531440, F₃, 21) (dual of [531440, 531272, 22]-code), using
        1 times truncation [i] based on linear OA(3¹⁶⁹, 531441, F₃, 22) (dual of [531441, 531272, 23]-code), using
        an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(179−21, 179, 185675)-Net over F₃ — Digital

Digital (158, 179, 185675)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁷⁹, 185675, F₃, 2, 21) (dual of [(185675, 2), 371171, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁷⁹, 265749, F₃, 2, 21) (dual of [(265749, 2), 531319, 22]-NRT-code), using
  - 1 step truncation [i] based on linear OOA(3¹⁸⁰, 265750, F₃, 2, 22) (dual of [(265750, 2), 531320, 23]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹⁸⁰, 531500, F₃, 22) (dual of [531500, 531320, 23]-code), using
      - constructionÂ X applied to C_e(21) âŠ‚ C_e(15) [i] based on
        linear OA(3¹⁶⁹, 531441, F₃, 22) (dual of [531441, 531272, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        linear OA(3¹²¹, 531441, F₃, 16) (dual of [531441, 531320, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
        linear OA(3¹¹, 59, F₃, 5) (dual of [59, 48, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹¹, 85, F₃, 5) (dual of [85, 74, 6]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(179−21, 179, large)-Net in Base 3 — Upper bound on s

There is no (158, 179, large)-net in base 3, because

19 times m-reduction [i] would yield (158, 160, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]