MinT - Best Known (113, 124, s)-Nets in Base 3

Best Known (113, 124, s)-Nets in Base 3

(113, 124, 1678005)-Net over F₃ — Constructive and digital

Digital (113, 124, 1678005)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 18, 285)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁸, 285, F₃, 5, 5) (dual of [(285, 5), 1407, 6]-NRT-code), using
    - appending kth column [i] based on linear OOA(3¹⁸, 285, F₃, 4, 5) (dual of [(285, 4), 1122, 6]-NRT-code), using
      - generalized (u,Â u+v)-construction [i] based on
        linear OOA(3¹, 95, F₃, 4, 1) (dual of [(95, 4), 379, 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, 95, F₃, 4, 2) (dual of [(95, 4), 375, 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(3¹², 95, F₃, 4, 5) (dual of [(95, 4), 368, 6]-NRT-code), using
        extracting embedded OOA [i] based on digital (7, 12, 95)-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]

(113, 124, 6876669)-Net over F₃ — Digital

Digital (113, 124, 6876669)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²⁴, 6876669, F₃, 11) (dual of [6876669, 6876545, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹²⁴, large, F₃, 11) (dual of [large, large−124, 12]-code), using
  - 18 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]

(113, 124, large)-Net in Base 3 — Upper bound on s

There is no (113, 124, large)-net in base 3, because

9 times m-reduction [i] would yield (113, 115, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (113, 124, s)-Nets in Base 3

(113, 124, 1678005)-Net over F3 — Constructive and digital

(113, 124, 6876669)-Net over F3 — Digital

(113, 124, large)-Net in Base 3 — Upper bound on s

(113, 124, 1678005)-Net over F₃ — Constructive and digital

(113, 124, 6876669)-Net over F₃ — Digital