Best Known (55−29, 55, s)-Nets in Base 27
(55−29, 55, 152)-Net over F27 — Constructive and digital
Digital (26, 55, 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, 35, 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
(55−29, 55, 172)-Net in Base 27 — Constructive
(26, 55, 172)-net in base 27, using
- 21 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
(55−29, 55, 323)-Net over F27 — Digital
Digital (26, 55, 323)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2755, 323, F27, 2, 29) (dual of [(323, 2), 591, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2755, 365, F27, 2, 29) (dual of [(365, 2), 675, 30]-NRT-code), using
- 271 times duplication [i] based on linear OOA(2754, 365, F27, 2, 29) (dual of [(365, 2), 676, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2754, 730, F27, 29) (dual of [730, 676, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(2754, 729, F27, 29) (dual of [729, 675, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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(270, 1, F27, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- OOA 2-folding [i] based on linear OA(2754, 730, F27, 29) (dual of [730, 676, 30]-code), using
- 271 times duplication [i] based on linear OOA(2754, 365, F27, 2, 29) (dual of [(365, 2), 676, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2755, 365, F27, 2, 29) (dual of [(365, 2), 675, 30]-NRT-code), using
(55−29, 55, 77164)-Net in Base 27 — Upper bound on s
There is no (26, 55, 77165)-net in base 27, because
- 1 times m-reduction [i] would yield (26, 54, 77165)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 196633 774847 156696 126624 516215 136945 328077 322011 378065 033632 293636 734464 149793 > 2754 [i]