MinT - Best Known (47, 47+44, s)-Nets in Base 27

Best Known (47, 47+44, s)-Nets in Base 27

(47, 47+44, 192)-Net over F₂₇ — Constructive and digital

Digital (47, 91, 192)-net over F₂₇, using

6 times m-reduction [i] based on digital (47, 97, 192)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 36, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
        an explicitly constructive algebraic function field [i]
  2. digital (11, 61, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(47, 47+44, 370)-Net in Base 27 — Constructive

(47, 91, 370)-net in base 27, using

t-expansion [i] based on (43, 91, 370)-net in base 27, using
- 17 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(47, 47+44, 722)-Net over F₂₇ — Digital

Digital (47, 91, 722)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁹¹, 722, F₂₇, 44) (dual of [722, 631, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(27⁹¹, 752, F₂₇, 44) (dual of [752, 661, 45]-code), using
  - constructionÂ X applied to C_e(43) âŠ‚ C_e(35) [i] based on
    1. linear OA(27⁸⁴, 729, F₂₇, 44) (dual of [729, 645, 45]-code), using an extension C_e(43) of the primitive narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [1,43], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 44 [i]
    2. linear OA(27⁶⁸, 729, F₂₇, 36) (dual of [729, 661, 37]-code), using an extension C_e(35) of the primitive narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [1,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [i]
    3. linear OA(27⁷, 23, F₂₇, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,27)), using
      - discarding factors / shortening the dual code based on linear OA(27⁷, 27, F₂₇, 7) (dual of [27, 20, 8]-code or 27-arc in PG(6,27)), using
        Reedâ€“Solomon code RS(20,27) [i]

(47, 47+44, 290070)-Net in Base 27 — Upper bound on s

There is no (47, 91, 290071)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 17952 115924 018074 089424 253469 086426 685246 878765 023036 868264 714402 037728 627749 536014 396330 476244 931453 634343 143515 485273 802124 706869 > 27⁹¹ [i]