MinT - Best Known (40−18, 40, s)-Nets in Base 49

Best Known (40−18, 40, s)-Nets in Base 49

(40−18, 40, 344)-Net over F₄₉ — Constructive and digital

Digital (22, 40, 344)-net over F₄₉, using

t-expansion [i] based on digital (21, 40, 344)-net over F₄₉, using
- net from sequence [i] based on digital (21, 343)-sequence over F₄₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 21 and N(F) ≥ 344, using
    - the Hermitian function field over F₄₉ [i]

(40−18, 40, 1860)-Net over F₄₉ — Digital

Digital (22, 40, 1860)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49⁴⁰, 1860, F₄₉, 18) (dual of [1860, 1820, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(49⁴⁰, 2418, F₄₉, 18) (dual of [2418, 2378, 19]-code), using
  - constructionÂ X applied to C_e(17) âŠ‚ C_e(11) [i] based on
    1. linear OA(49³⁵, 2401, F₄₉, 18) (dual of [2401, 2366, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
    2. linear OA(49²³, 2401, F₄₉, 12) (dual of [2401, 2378, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
    3. linear OA(49⁵, 17, F₄₉, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,49)), using
      - discarding factors / shortening the dual code based on linear OA(49⁵, 49, F₄₉, 5) (dual of [49, 44, 6]-code or 49-arc in PG(4,49)), using
        Reedâ€“Solomon code RS(44,49) [i]

(40−18, 40, 2808614)-Net in Base 49 — Upper bound on s

There is no (22, 40, 2808615)-net in base 49, because

the generalized Rao bound for nets shows that 49^mÂ â‰¥ 40 536274 267493 203130 821661 096227 015950 008278 938467 023397 508856 545745 > 49⁴⁰ [i]