MinT - Best Known (84−41, 84, s)-Nets in Base 27

Best Known (84−41, 84, s)-Nets in Base 27

(84−41, 84, 192)-Net over F₂₇ — Constructive and digital

Digital (43, 84, 192)-net over F₂₇, using

1 times m-reduction [i] based on digital (43, 85, 192)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 32, 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, 53, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(84−41, 84, 370)-Net in Base 27 — Constructive

(43, 84, 370)-net in base 27, using

24 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]

(84−41, 84, 641)-Net over F₂₇ — Digital

Digital (43, 84, 641)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁸⁴, 641, F₂₇, 41) (dual of [641, 557, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(27⁸⁴, 741, F₂₇, 41) (dual of [741, 657, 42]-code), using
  - constructionÂ X applied to C([0,20]) âŠ‚ C([0,18]) [i] based on
    1. linear OA(27⁸¹, 730, F₂₇, 41) (dual of [730, 649, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 27⁴âˆ’1, defining interval IÂ =Â [0,20], and minimum distance dÂ â‰¥ |{−20,−19,…,20}|+1Â =Â 42 (BCH-bound) [i]
    2. linear OA(27⁷³, 730, F₂₇, 37) (dual of [730, 657, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 27⁴âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
    3. linear OA(27³, 11, F₂₇, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,27) or 11-cap in PG(2,27)), using
      - discarding factors / shortening the dual code based on linear OA(27³, 27, F₂₇, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
        Reedâ€“Solomon code RS(24,27) [i]

(84−41, 84, 278276)-Net in Base 27 — Upper bound on s

There is no (43, 84, 278277)-net in base 27, because

1 times m-reduction [i] would yield (43, 83, 278277)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 63563 693287 982487 686222 779119 877089 876019 157826 994719 657972 694189 830674 738917 020298 577890 375489 342514 345081 408889 174321 > 27⁸³ [i]