MinT - Best Known (114−27, 114, s)-Nets in Base 3

Best Known (114−27, 114, s)-Nets in Base 3

(114−27, 114, 400)-Net over F₃ — Constructive and digital

Digital (87, 114, 400)-net over F₃, using

3² times duplication [i] based on digital (85, 112, 400)-net over F₃, using
- trace code for nets [i] based on digital (1, 28, 100)-net over F₈₁, using
  - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
      - a function field by SÃ©mirat [i]

(114−27, 114, 708)-Net over F₃ — Digital

Digital (87, 114, 708)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹¹⁴, 708, F₃, 27) (dual of [708, 594, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹¹⁴, 749, F₃, 27) (dual of [749, 635, 28]-code), using
  - constructionÂ X applied to C_e(27) âŠ‚ C_e(22) [i] based on
    1. linear OA(3¹⁰⁹, 729, F₃, 28) (dual of [729, 620, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    2. linear OA(3⁹¹, 729, F₃, 23) (dual of [729, 638, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    3. linear OA(3⁵, 20, F₃, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
      - doubling the cap [i] based on linear OA(3⁴, 10, F₃, 3) (dual of [10, 6, 4]-code or 10-cap in PG(3,3)), using
        ovoid in PG(3,Â 3) [i]

(114−27, 114, 39767)-Net in Base 3 — Upper bound on s

There is no (87, 114, 39768)-net in base 3, because

1 times m-reduction [i] would yield (87, 113, 39768)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 821906 623954 165515 661233 765192 083748 821502 287351 694257 > 3¹¹³ [i]