MinT - Best Known (178−23, 178, s)-Nets in Base 3

Best Known (178−23, 178, s)-Nets in Base 3

(178−23, 178, 16112)-Net over F₃ — Constructive and digital

Digital (155, 178, 16112)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 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 (143, 166, 16105)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁶⁶, 16105, F₃, 23, 23) (dual of [(16105, 23), 370249, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(3¹⁶⁶, 177156, F₃, 23) (dual of [177156, 176990, 24]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁶⁶, 177158, F₃, 23) (dual of [177158, 176992, 24]-code), using
        constructionÂ X applied to C_e(22) âŠ‚ C_e(21) [i] based on
        linear OA(3¹⁶⁶, 177147, F₃, 23) (dual of [177147, 176981, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
        linear OA(3¹⁵⁵, 177147, F₃, 22) (dual of [177147, 176992, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [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]

(178−23, 178, 65587)-Net over F₃ — Digital

Digital (155, 178, 65587)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁷⁸, 65587, F₃, 2, 23) (dual of [(65587, 2), 130996, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁷⁸, 88601, F₃, 2, 23) (dual of [(88601, 2), 177024, 24]-NRT-code), using
  - 3¹ times duplication [i] based on linear OOA(3¹⁷⁷, 88601, F₃, 2, 23) (dual of [(88601, 2), 177025, 24]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹⁷⁷, 177202, F₃, 23) (dual of [177202, 177025, 24]-code), using
      - constructionÂ X applied to C_e(22) âŠ‚ C_e(16) [i] based on
        linear OA(3¹⁶⁶, 177147, F₃, 23) (dual of [177147, 176981, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
        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(3¹¹, 55, F₃, 5) (dual of [55, 44, 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]

(178−23, 178, large)-Net in Base 3 — Upper bound on s

There is no (155, 178, large)-net in base 3, because

21 times m-reduction [i] would yield (155, 157, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (178−23, 178, s)-Nets in Base 3

(178−23, 178, 16112)-Net over F3 — Constructive and digital

(178−23, 178, 65587)-Net over F3 — Digital

(178−23, 178, large)-Net in Base 3 — Upper bound on s

(178−23, 178, 16112)-Net over F₃ — Constructive and digital

(178−23, 178, 65587)-Net over F₃ — Digital