MinT - Best Known (27−14, 27, s)-Nets in Base 49

Best Known (27−14, 27, s)-Nets in Base 49

(27−14, 27, 343)-Net over F₄₉ — Constructive and digital

Digital (13, 27, 343)-net over F₄₉, using

net defined by OOA [i] based on linear OOA(49²⁷, 343, F₄₉, 14, 14) (dual of [(343, 14), 4775, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(49²⁷, 2401, F₄₉, 14) (dual of [2401, 2374, 15]-code), using
  - an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(27−14, 27, 801)-Net over F₄₉ — Digital

Digital (13, 27, 801)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49²⁷, 801, F₄₉, 3, 14) (dual of [(801, 3), 2376, 15]-NRT-code), using
- OOA 3-folding [i] based on linear OA(49²⁷, 2403, F₄₉, 14) (dual of [2403, 2376, 15]-code), using
  - constructionÂ X applied to C_e(13) âŠ‚ C_e(12) [i] based on
    1. linear OA(49²⁷, 2401, F₄₉, 14) (dual of [2401, 2374, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
    2. linear OA(49²⁵, 2401, F₄₉, 13) (dual of [2401, 2376, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    3. linear OA(49⁰, 2, F₄₉, 0) (dual of [2, 2, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(49⁰, s, F₄₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(27−14, 27, 232808)-Net in Base 49 — Upper bound on s

There is no (13, 27, 232809)-net in base 49, because

the generalized Rao bound for nets shows that 49^mÂ â‰¥ 4318 148378 803576 804865 023920 160183 276313 309905 > 49²⁷ [i]