MinT - Best Known (13, 17, s)-Nets in Base 49

Best Known (13, 17, s)-Nets in Base 49

(13, 17, 4194301)-Net over F₄₉ — Constructive and digital

Digital (13, 17, 4194301)-net over F₄₉, using

49¹ times duplication [i] based on digital (12, 16, 4194301)-net over F₄₉, using
- net defined by OOA [i] based on linear OOA(49¹⁶, 4194301, F₄₉, 4, 4) (dual of [(4194301, 4), 16777188, 5]-NRT-code), using
  - appending kth column [i] based on linear OOA(49¹⁶, 4194301, F₄₉, 3, 4) (dual of [(4194301, 3), 12582887, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(49¹⁶, 8388602, F₄₉, 4) (dual of [8388602, 8388586, 5]-code), using
      - discarding factors / shortening the dual code based on linear OA(49¹⁶, large, F₄₉, 4) (dual of [large, large−16, 5]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 25679568Â | 49⁵âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

(13, 17, large)-Net over F₄₉ — Digital

Digital (13, 17, large)-net over F₄₉, using

49² times duplication [i] based on digital (11, 15, large)-net over F₄₉, using
- net defined by OOA [i] based on linear OOA(49¹⁵, large, F₄₉, 4, 4), using
  - appending kth column [i] based on linear OOA(49¹⁵, large, F₄₉, 3, 4), using
    - Gilbertâ€“VarÅ¡amov bound [i]

(13, 17, large)-Net in Base 49 — Upper bound on s

There is no (13, 17, large)-net in base 49, because

2 times m-reduction [i] would yield (13, 15, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]