MinT - Best Known (122−11, 122, s)-Nets in Base 3

Best Known (122−11, 122, s)-Nets in Base 3

(122−11, 122, 1677877)-Net over F₃ — Constructive and digital

Digital (111, 122, 1677877)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (11, 16, 157)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁶, 157, F₃, 5, 5) (dual of [(157, 5), 769, 6]-NRT-code), using
    - appending kth column [i] based on linear OOA(3¹⁶, 157, F₃, 4, 5) (dual of [(157, 4), 612, 6]-NRT-code), using
      - generalized (u,Â u+v)-construction [i] based on
        linear OOA(3¹, 40, F₃, 4, 1) (dual of [(40, 4), 159, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(3¹, s, F₃, 4, 1) with arbitrarily large s, using
        appending 3 arbitrary columns [i] based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
        linear OOA(3⁴, 40, F₃, 4, 2) (dual of [(40, 4), 156, 3]-NRT-code), using
        appending 2 arbitrary columns [i] based on linear OOA(3⁴, 40, F₃, 2, 2) (dual of [(40, 2), 76, 3]-NRT-code), using
        appending kth column [i] based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]
        linear OOA(3¹¹, 77, F₃, 4, 5) (dual of [(77, 4), 297, 6]-NRT-code), using
        extracting embedded OOA [i] based on digital (6, 11, 77)-net over F₃, using
        linear code embedding found in a computer search [i]
2. digital (95, 106, 1677720)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁰⁶, 1677720, F₃, 11, 11) (dual of [(1677720, 11), 18454814, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(3¹⁰⁶, 8388601, F₃, 11) (dual of [8388601, 8388495, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁰⁶, large, F₃, 11) (dual of [large, large−106, 12]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(122−11, 122, 5387050)-Net over F₃ — Digital

Digital (111, 122, 5387050)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²², 5387050, F₃, 11) (dual of [5387050, 5386928, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹²², large, F₃, 11) (dual of [large, large−122, 12]-code), using
  - 16 times code embedding in larger space [i] based on linear OA(3¹⁰⁶, large, F₃, 11) (dual of [large, large−106, 12]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(122−11, 122, large)-Net in Base 3 — Upper bound on s

There is no (111, 122, large)-net in base 3, because

9 times m-reduction [i] would yield (111, 113, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (122−11, 122, s)-Nets in Base 3

(122−11, 122, 1677877)-Net over F3 — Constructive and digital

(122−11, 122, 5387050)-Net over F3 — Digital

(122−11, 122, large)-Net in Base 3 — Upper bound on s

(122−11, 122, 1677877)-Net over F₃ — Constructive and digital

(122−11, 122, 5387050)-Net over F₃ — Digital