Best Known (54−28, 54, s)-Nets in Base 27
(54−28, 54, 152)-Net over F27 — Constructive and digital
Digital (26, 54, 152)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (6, 20, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 6 and N(F) ≥ 76, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- digital (6, 34, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27 (see above)
- digital (6, 20, 76)-net over F27, using
(54−28, 54, 172)-Net in Base 27 — Constructive
(26, 54, 172)-net in base 27, using
- 22 times m-reduction [i] based on (26, 76, 172)-net in base 27, using
- base change [i] based on digital (7, 57, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- base change [i] based on digital (7, 57, 172)-net over F81, using
(54−28, 54, 359)-Net over F27 — Digital
Digital (26, 54, 359)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2754, 359, F27, 2, 28) (dual of [(359, 2), 664, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2754, 366, F27, 2, 28) (dual of [(366, 2), 678, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2754, 732, F27, 28) (dual of [732, 678, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- linear OA(2753, 729, F27, 28) (dual of [729, 676, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(2751, 729, F27, 26) (dual of [729, 678, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(271, 3, F27, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- OOA 2-folding [i] based on linear OA(2754, 732, F27, 28) (dual of [732, 678, 29]-code), using
- discarding factors / shortening the dual code based on linear OOA(2754, 366, F27, 2, 28) (dual of [(366, 2), 678, 29]-NRT-code), using
(54−28, 54, 77164)-Net in Base 27 — Upper bound on s
There is no (26, 54, 77165)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 196633 774847 156696 126624 516215 136945 328077 322011 378065 033632 293636 734464 149793 > 2754 [i]