MinT - Best Known (51−28, 51, s)-Nets in Base 49

Best Known (51−28, 51, s)-Nets in Base 49

(51−28, 51, 344)-Net over F₄₉ — Constructive and digital

Digital (23, 51, 344)-net over F₄₉, using

t-expansion [i] based on digital (21, 51, 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]

(51−28, 51, 370)-Net over F₄₉ — Digital

Digital (23, 51, 370)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49⁵¹, 370, F₄₉, 28) (dual of [370, 319, 29]-code), using
- 23 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 22 times 0) [i] based on linear OA(49⁵⁰, 346, F₄₉, 28) (dual of [346, 296, 29]-code), using
  - constructionÂ X applied to AG(F,314P) âŠ‚ AG(F,316P) [i] based on
    1. linear OA(49⁴⁹, 343, F₄₉, 28) (dual of [343, 294, 29]-code), using algebraic-geometric code AG(F,314P) [i] based on function field F/F₄₉ with g(F) = 21 and N(F) ≥ 344, using
      - the Hermitian function field over F₄₉ [i]
    2. linear OA(49⁴⁷, 343, F₄₉, 26) (dual of [343, 296, 27]-code), using algebraic-geometric code AG(F,316P) [i] based on function field F/F₄₉ with g(F) = 21 and N(F) ≥ 344 (see above)
    3. linear OA(49¹, 3, F₄₉, 1) (dual of [3, 2, 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]

(51−28, 51, 180860)-Net in Base 49 — Upper bound on s

There is no (23, 51, 180861)-net in base 49, because

the generalized Rao bound for nets shows that 49^mÂ â‰¥ 158 498907 755671 023204 812318 995807 429049 658266 682190 721466 390709 007298 714085 552962 085153 > 49⁵¹ [i]