MinT - Best Known (21, 21+16, s)-Nets in Base 49

Best Known (21, 21+16, s)-Nets in Base 49

(21, 21+16, 344)-Net over F₄₉ — Constructive and digital

Digital (21, 37, 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+16, 2518)-Net over F₄₉ — Digital

Digital (21, 37, 2518)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49³⁷, 2518, F₄₉, 16) (dual of [2518, 2481, 17]-code), using
- 107 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 4 times 0, 1, 21 times 0, 1, 79 times 0) [i] based on linear OA(49³², 2406, F₄₉, 16) (dual of [2406, 2374, 17]-code), using
  - constructionÂ X applied to C_e(15) âŠ‚ C_e(13) [i] based on
    1. linear OA(49³¹, 2401, F₄₉, 16) (dual of [2401, 2370, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    2. 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]
    3. linear OA(49¹, 5, F₄₉, 1) (dual of [5, 4, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(49¹, s, F₄₉, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(21, 21+16, 5147607)-Net in Base 49 — Upper bound on s

There is no (21, 37, 5147608)-net in base 49, because

the generalized Rao bound for nets shows that 49^mÂ â‰¥ 344 552644 531496 807879 594024 131711 258471 494397 252227 941474 079745 > 49³⁷ [i]