Best Known (39−19, 39, s)-Nets in Base 27
(39−19, 39, 146)-Net over F27 — Constructive and digital
Digital (20, 39, 146)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 13, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- digital (7, 26, 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 (4, 13, 64)-net over F27, using
(39−19, 39, 182)-Net in Base 27 — Constructive
(20, 39, 182)-net in base 27, using
- (u, u+v)-construction [i] based on
- (3, 12, 82)-net in base 27, using
- base change [i] based on digital (0, 9, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- base change [i] based on digital (0, 9, 82)-net over F81, using
- (8, 27, 100)-net in base 27, using
- 1 times m-reduction [i] based on (8, 28, 100)-net in base 27, using
- base change [i] based on digital (1, 21, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- base change [i] based on digital (1, 21, 100)-net over F81, using
- 1 times m-reduction [i] based on (8, 28, 100)-net in base 27, using
- (3, 12, 82)-net in base 27, using
(39−19, 39, 429)-Net over F27 — Digital
Digital (20, 39, 429)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2739, 429, F27, 19) (dual of [429, 390, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2739, 737, F27, 19) (dual of [737, 698, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(2737, 729, F27, 19) (dual of [729, 692, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2731, 729, F27, 16) (dual of [729, 698, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(272, 8, F27, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(2739, 737, F27, 19) (dual of [737, 698, 20]-code), using
(39−19, 39, 176320)-Net in Base 27 — Upper bound on s
There is no (20, 39, 176321)-net in base 27, because
- 1 times m-reduction [i] would yield (20, 38, 176321)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 2 465073 109657 109677 961628 110706 541320 817040 488200 183611 > 2738 [i]