MinT - Best Known (200−21, 200, s)-Nets in Base 3

Best Known (200−21, 200, s)-Nets in Base 3

(200−21, 200, 478300)-Net over F₃ — Constructive and digital

Digital (179, 200, 478300)-net over F₃, using

3¹ times duplication [i] based on digital (178, 199, 478300)-net over F₃, using
- net defined by OOA [i] based on linear OOA(3¹⁹⁹, 478300, F₃, 21, 21) (dual of [(478300, 21), 10044101, 22]-NRT-code), using
  - OOA 10-folding and stacking with additional row [i] based on linear OA(3¹⁹⁹, 4783001, F₃, 21) (dual of [4783001, 4782802, 22]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(3¹⁹⁸, 4783000, F₃, 21) (dual of [4783000, 4782802, 22]-code), using
      - constructionÂ X4 applied to C([0,10]) âŠ‚ C([0,9]) [i] based on
        linear OA(3¹⁹⁷, 4782970, F₃, 21) (dual of [4782970, 4782773, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970Â | 3²⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(3¹⁶⁹, 4782970, F₃, 19) (dual of [4782970, 4782801, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970Â | 3²⁸âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
        linear OA(3²⁹, 30, F₃, 29) (dual of [30, 1, 30]-code or 30-arc in PG(28,3)), using
        dual of repetition code with length 30 [i]
        linear OA(3¹, 30, F₃, 1) (dual of [30, 29, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(200−21, 200, 1213117)-Net over F₃ — Digital

Digital (179, 200, 1213117)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²⁰⁰, 1213117, F₃, 3, 21) (dual of [(1213117, 3), 3639151, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²⁰⁰, 1594334, F₃, 3, 21) (dual of [(1594334, 3), 4782802, 22]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(3²⁰⁰, 4783002, F₃, 21) (dual of [4783002, 4782802, 22]-code), using
    - 2 times code embedding in larger space [i] based on linear OA(3¹⁹⁸, 4783000, F₃, 21) (dual of [4783000, 4782802, 22]-code), using
      - constructionÂ X4 applied to C([0,10]) âŠ‚ C([0,9]) [i] based on
        linear OA(3¹⁹⁷, 4782970, F₃, 21) (dual of [4782970, 4782773, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970Â | 3²⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(3¹⁶⁹, 4782970, F₃, 19) (dual of [4782970, 4782801, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970Â | 3²⁸âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
        linear OA(3²⁹, 30, F₃, 29) (dual of [30, 1, 30]-code or 30-arc in PG(28,3)), using
        dual of repetition code with length 30 [i]
        linear OA(3¹, 30, F₃, 1) (dual of [30, 29, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(200−21, 200, large)-Net in Base 3 — Upper bound on s

There is no (179, 200, large)-net in base 3, because

19 times m-reduction [i] would yield (179, 181, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (200−21, 200, s)-Nets in Base 3

(200−21, 200, 478300)-Net over F3 — Constructive and digital

(200−21, 200, 1213117)-Net over F3 — Digital

(200−21, 200, large)-Net in Base 3 — Upper bound on s

(200−21, 200, 478300)-Net over F₃ — Constructive and digital

(200−21, 200, 1213117)-Net over F₃ — Digital