MinT - Best Known (200−21, 200, s)-Nets in Base 2

Best Known (200−21, 200, s)-Nets in Base 2

(200−21, 200, 52433)-Net over F₂ — Constructive and digital

Digital (179, 200, 52433)-net over F₂, using

2² times duplication [i] based on digital (177, 198, 52433)-net over F₂, using
- net defined by OOA [i] based on linear OOA(2¹⁹⁸, 52433, F₂, 21, 21) (dual of [(52433, 21), 1100895, 22]-NRT-code), using
  - OOA 10-folding and stacking with additional row [i] based on linear OA(2¹⁹⁸, 524331, F₂, 21) (dual of [524331, 524133, 22]-code), using
    - discarding factors / shortening the dual code based on linear OA(2¹⁹⁸, 524334, F₂, 21) (dual of [524334, 524136, 22]-code), using
      - constructionÂ X applied to C([0,10]) âŠ‚ C([0,8]) [i] based on
        linear OA(2¹⁹¹, 524289, F₂, 21) (dual of [524289, 524098, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 524289Â | 2³⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(2¹⁵³, 524289, F₂, 17) (dual of [524289, 524136, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 524289Â | 2³⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
        linear OA(2⁷, 45, F₂, 3) (dual of [45, 38, 4]-code or 45-cap in PG(6,2)), using
        discarding factors / shortening the dual code based on linear OA(2⁷, 63, F₂, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 2⁶âˆ’1, defining interval IÂ =Â [0,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]

(200−21, 200, 87389)-Net over F₂ — Digital

Digital (179, 200, 87389)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁰⁰, 87389, F₂, 6, 21) (dual of [(87389, 6), 524134, 22]-NRT-code), using
- 2² times duplication [i] based on linear OOA(2¹⁹⁸, 87389, F₂, 6, 21) (dual of [(87389, 6), 524136, 22]-NRT-code), using
  - OOA 6-folding [i] based on linear OA(2¹⁹⁸, 524334, F₂, 21) (dual of [524334, 524136, 22]-code), using
    - constructionÂ X applied to C([0,10]) âŠ‚ C([0,8]) [i] based on
      1. linear OA(2¹⁹¹, 524289, F₂, 21) (dual of [524289, 524098, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 524289Â | 2³⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
      2. linear OA(2¹⁵³, 524289, F₂, 17) (dual of [524289, 524136, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 524289Â | 2³⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
      3. linear OA(2⁷, 45, F₂, 3) (dual of [45, 38, 4]-code or 45-cap in PG(6,2)), using
        discarding factors / shortening the dual code based on linear OA(2⁷, 63, F₂, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 2⁶âˆ’1, defining interval IÂ =Â [0,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]

(200−21, 200, 4430694)-Net in Base 2 — Upper bound on s

There is no (179, 200, 4430695)-net in base 2, because

1 times m-reduction [i] would yield (179, 199, 4430695)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 803470 120919 764146 785535 354805 457908 515556 278087 904845 330884 > 2¹⁹⁹ [i]

Best Known (200−21, 200, s)-Nets in Base 2

(200−21, 200, 52433)-Net over F2 — Constructive and digital

(200−21, 200, 87389)-Net over F2 — Digital

(200−21, 200, 4430694)-Net in Base 2 — Upper bound on s

(200−21, 200, 52433)-Net over F₂ — Constructive and digital

(200−21, 200, 87389)-Net over F₂ — Digital