MinT - Best Known (42, 65, s)-Nets in Base 27

Best Known (42, 65, s)-Nets in Base 27

(42, 65, 336)-Net over F₂₇ — Constructive and digital

Digital (42, 65, 336)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 1, 28)-net over F₂₇, using
  - s-reduction based on digital (0, 1, s)-net over F₂₇ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
2. digital (0, 2, 28)-net over F₂₇, using
  - kÂ =Â 2 construction [i]
3. digital (0, 2, 28)-net over F₂₇ (see above)
4. digital (0, 2, 28)-net over F₂₇ (see above)
5. digital (0, 2, 28)-net over F₂₇ (see above)
6. digital (0, 3, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
      - the rational function field F₂₇(x) [i]
    - Niederreiter sequence [i]
7. digital (0, 3, 28)-net over F₂₇ (see above)
8. digital (0, 4, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
9. digital (0, 5, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
10. digital (0, 7, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
11. digital (0, 11, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
12. digital (0, 23, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)

(42, 65, 597)-Net in Base 27 — Constructive

(42, 65, 597)-net in base 27, using

27¹ times duplication [i] based on (41, 64, 597)-net in base 27, using
- base change [i] based on digital (25, 48, 597)-net over F₈₁, using
  - 81¹ times duplication [i] based on digital (24, 47, 597)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81⁴⁷, 597, F₈₁, 23, 23) (dual of [(597, 23), 13684, 24]-NRT-code), using
      - OOA 11-folding and stacking with additional row [i] based on linear OA(81⁴⁷, 6568, F₈₁, 23) (dual of [6568, 6521, 24]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁴⁷, 6569, F₈₁, 23) (dual of [6569, 6522, 24]-code), using
        constructionÂ X applied to C_e(22) âŠ‚ C_e(19) [i] based on
        linear OA(81⁴⁵, 6561, F₈₁, 23) (dual of [6561, 6516, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
        linear OA(81³⁹, 6561, F₈₁, 20) (dual of [6561, 6522, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(81², 8, F₈₁, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,81)), using
        discarding factors / shortening the dual code based on linear OA(81², 81, F₈₁, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
        Reedâ€“Solomon code RS(79,81) [i]

(42, 65, 5912)-Net over F₂₇ — Digital

Digital (42, 65, 5912)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 65, large)-Net in Base 27 — Upper bound on s

There is no (42, 65, large)-net in base 27, because

21 times m-reduction [i] would yield (42, 44, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (42, 65, s)-Nets in Base 27

(42, 65, 336)-Net over F27 — Constructive and digital

(42, 65, 597)-Net in Base 27 — Constructive

(42, 65, 5912)-Net over F27 — Digital

(42, 65, large)-Net in Base 27 — Upper bound on s

(42, 65, 336)-Net over F₂₇ — Constructive and digital

(42, 65, 5912)-Net over F₂₇ — Digital