MinT - Best Known (21, 21+18, s)-Nets in Base 49

Best Known (21, 21+18, s)-Nets in Base 49

(21, 21+18, 344)-Net over F₄₉ — Constructive and digital

Digital (21, 39, 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]

(21, 21+18, 1457)-Net over F₄₉ — Digital

Digital (21, 39, 1457)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49³⁹, 1457, F₄₉, 18) (dual of [1457, 1418, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(49³⁹, 2415, F₄₉, 18) (dual of [2415, 2376, 19]-code), using
  - constructionÂ X applied to C_e(17) âŠ‚ C_e(12) [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(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(49⁴, 14, F₄₉, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,49)), using
      - discarding factors / shortening the dual code based on linear OA(49⁴, 49, F₄₉, 4) (dual of [49, 45, 5]-code or 49-arc in PG(3,49)), using
        Reedâ€“Solomon code RS(45,49) [i]

(21, 21+18, 1822603)-Net in Base 49 — Upper bound on s

There is no (21, 39, 1822604)-net in base 49, because

the generalized Rao bound for nets shows that 49^mÂ â‰¥ 827271 295210 314943 349124 430601 311646 181007 362856 029711 610562 671681 > 49³⁹ [i]