MinT - Best Known (23, 42, s)-Nets in Base 49

Best Known (23, 42, s)-Nets in Base 49

(23, 42, 344)-Net over F₄₉ — Constructive and digital

Digital (23, 42, 344)-net over F₄₉, using

t-expansion [i] based on digital (21, 42, 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]

(23, 42, 1775)-Net over F₄₉ — Digital

Digital (23, 42, 1775)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49⁴², 1775, F₄₉, 19) (dual of [1775, 1733, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(49⁴², 2419, F₄₉, 19) (dual of [2419, 2377, 20]-code), using
  - constructionÂ X applied to C([0,9]) âŠ‚ C([0,6]) [i] based on
    1. linear OA(49³⁷, 2402, F₄₉, 19) (dual of [2402, 2365, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402Â | 49⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
    2. linear OA(49²⁵, 2402, F₄₉, 13) (dual of [2402, 2377, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402Â | 49⁴âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [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]

(23, 42, 4328046)-Net in Base 49 — Upper bound on s

There is no (23, 42, 4328047)-net in base 49, because

1 times m-reduction [i] would yield (23, 41, 4328047)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 1986 276022 197900 638684 086147 683029 720501 718081 654367 924531 196662 037329 > 49⁴¹ [i]