MinT - Best Known (213−13, 213, s)-Nets in Base 3

Best Known (213−13, 213, s)-Nets in Base 3

(213−13, 213, 5592453)-Net over F₃ — Constructive and digital

Digital (200, 213, 5592453)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (11, 17, 53)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁷, 53, F₃, 6, 6) (dual of [(53, 6), 301, 7]-NRT-code), using
    - appending kth column [i] based on linear OOA(3¹⁷, 53, F₃, 5, 6) (dual of [(53, 5), 248, 7]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(3⁵, 20, F₃, 5, 3) (dual of [(20, 5), 95, 4]-NRT-code), using
        appending 2 arbitrary columns [i] based on linear OOA(3⁵, 20, F₃, 3, 3) (dual of [(20, 3), 55, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(3⁵, 20, F₃, 2, 3) (dual of [(20, 2), 35, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(3⁵, 20, F₃, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
        doubling the cap [i] based on linear OA(3⁴, 10, F₃, 3) (dual of [10, 6, 4]-code or 10-cap in PG(3,3)), using
        ovoid in PG(3,Â 3) [i]
        linear OOA(3¹², 33, F₃, 5, 6) (dual of [(33, 5), 153, 7]-NRT-code), using
        extracting embedded OOA [i] based on digital (6, 12, 33)-net over F₃, using
        linear code embedding found in a computer search [i]
2. digital (183, 196, 5592400)-net over F₃, using
  - trace code for nets [i] based on digital (85, 98, 2796200)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9⁹⁸, 2796200, F₉, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OOA(9⁹⁸, 8388601, F₉, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(9⁹⁸, 8388602, F₉, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
        trace code [i] based on linear OOA(81⁴⁹, 4194301, F₈₁, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(81⁴⁹, 8388602, F₈₁, 13) (dual of [8388602, 8388553, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁴⁹, large, F₈₁, 13) (dual of [large, large−49, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(213−13, 213, large)-Net over F₃ — Digital

Digital (200, 213, large)-net over F₃, using

t-expansion [i] based on digital (198, 213, large)-net over F₃, using
- 3 times m-reduction [i] based on digital (198, 216, large)-net over F₃, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²¹⁶, large, F₃, 18) (dual of [large, large−216, 19]-code), using
    - 36 times code embedding in larger space [i] based on linear OA(3¹⁸⁰, large, F₃, 18) (dual of [large, large−180, 19]-code), using
      - the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(213−13, 213, large)-Net in Base 3 — Upper bound on s

There is no (200, 213, large)-net in base 3, because

11 times m-reduction [i] would yield (200, 202, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (213−13, 213, s)-Nets in Base 3

(213−13, 213, 5592453)-Net over F3 — Constructive and digital

(213−13, 213, large)-Net over F3 — Digital

(213−13, 213, large)-Net in Base 3 — Upper bound on s

(213−13, 213, 5592453)-Net over F₃ — Constructive and digital

(213−13, 213, large)-Net over F₃ — Digital