Best Known (28, 28+25, s)-Nets in Base 27
(28, 28+25, 170)-Net over F27 — Constructive and digital
Digital (28, 53, 170)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 19, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 7 and N(F) ≥ 82, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- digital (9, 34, 88)-net over F27, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 9 and N(F) ≥ 88, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- digital (7, 19, 82)-net over F27, using
(28, 28+25, 224)-Net in Base 27 — Constructive
(28, 53, 224)-net in base 27, using
- 7 times m-reduction [i] based on (28, 60, 224)-net in base 27, using
- base change [i] based on digital (13, 45, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- base change [i] based on digital (13, 45, 224)-net over F81, using
(28, 28+25, 614)-Net over F27 — Digital
Digital (28, 53, 614)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2753, 614, F27, 25) (dual of [614, 561, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2753, 743, F27, 25) (dual of [743, 690, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(2749, 729, F27, 25) (dual of [729, 680, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2739, 729, F27, 20) (dual of [729, 690, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(274, 14, F27, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,27)), using
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- Reed–Solomon code RS(23,27) [i]
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(2753, 743, F27, 25) (dual of [743, 690, 26]-code), using
(28, 28+25, 324307)-Net in Base 27 — Upper bound on s
There is no (28, 53, 324308)-net in base 27, because
- 1 times m-reduction [i] would yield (28, 52, 324308)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 269 731056 662135 201037 013510 408751 075961 843403 061168 754975 736344 497556 848769 > 2752 [i]