MinT - Best Known (38−10, 38, s)-Nets in Base 3

Best Known (38−10, 38, s)-Nets in Base 3

(38−10, 38, 164)-Net over F₃ — Constructive and digital

Digital (28, 38, 164)-net over F₃, using

trace code for nets [i] based on digital (9, 19, 82)-net over F₉, using
- base reduction for projective spaces (embedding PG(9,81) in PG(18,9)) for nets [i] based on digital (0, 10, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
      - the rational function field F₈₁(x) [i]
    - Niederreiter sequence [i]

(38−10, 38, 368)-Net over F₃ — Digital

Digital (28, 38, 368)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3³⁸, 368, F₃, 2, 10) (dual of [(368, 2), 698, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3³⁸, 736, F₃, 10) (dual of [736, 698, 11]-code), using
  - constructionÂ X applied to C_e(9) âŠ‚ C_e(7) [i] based on
    1. linear OA(3³⁷, 729, F₃, 10) (dual of [729, 692, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
    2. linear OA(3³¹, 729, F₃, 8) (dual of [729, 698, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
    3. linear OA(3¹, 7, F₃, 1) (dual of [7, 6, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(38−10, 38, 5502)-Net in Base 3 — Upper bound on s

There is no (28, 38, 5503)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 351265 389270 417143 > 3³⁸ [i]