MinT - Best Known (123−11, 123, s)-Nets in Base 3

Best Known (123−11, 123, s)-Nets in Base 3

(123−11, 123, 1677951)-Net over F₃ — Constructive and digital

Digital (112, 123, 1677951)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (12, 17, 231)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁷, 231, F₃, 5, 5) (dual of [(231, 5), 1138, 6]-NRT-code), using
    - appending kth column [i] based on linear OOA(3¹⁷, 231, F₃, 4, 5) (dual of [(231, 4), 907, 6]-NRT-code), using
      - generalized (u,Â u+v)-construction [i] based on
        linear OOA(3¹, 77, F₃, 4, 1) (dual of [(77, 4), 307, 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(3⁵, 77, F₃, 4, 2) (dual of [(77, 4), 303, 3]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(3⁵, 121, F₃, 4, 2) (dual of [(121, 4), 479, 3]-NRT-code), using
        appending 2 arbitrary columns [i] based on linear OOA(3⁵, 121, F₃, 2, 2) (dual of [(121, 2), 237, 3]-NRT-code), using
        appending kth column [i] based on linear OA(3⁵, 121, F₃, 2) (dual of [121, 116, 3]-code), using
        Hamming code H(5,3) [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]

(123−11, 123, 6086457)-Net over F₃ — Digital

Digital (112, 123, 6086457)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²³, 6086457, F₃, 11) (dual of [6086457, 6086334, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹²³, large, F₃, 11) (dual of [large, large−123, 12]-code), using
  - 17 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]

(123−11, 123, large)-Net in Base 3 — Upper bound on s

There is no (112, 123, large)-net in base 3, because

9 times m-reduction [i] would yield (112, 114, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (123−11, 123, s)-Nets in Base 3

(123−11, 123, 1677951)-Net over F3 — Constructive and digital

(123−11, 123, 6086457)-Net over F3 — Digital

(123−11, 123, large)-Net in Base 3 — Upper bound on s

(123−11, 123, 1677951)-Net over F₃ — Constructive and digital

(123−11, 123, 6086457)-Net over F₃ — Digital