Best Known (28, 28+21, s)-Nets in Base 27
(28, 28+21, 184)-Net over F27 — Constructive and digital
Digital (28, 49, 184)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (3, 10, 56)-net over F27, using
- (u, u+v)-construction [i] based on
- 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]
- Niederreiter–Xing sequence construction 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]
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- digital (0, 7, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 3, 28)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 14, 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 (4, 25, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27 (see above)
- digital (3, 10, 56)-net over F27, using
(28, 28+21, 250)-Net in Base 27 — Constructive
(28, 49, 250)-net in base 27, using
- 271 times duplication [i] based on (27, 48, 250)-net in base 27, using
- base change [i] based on digital (15, 36, 250)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 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
- digital (4, 25, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- digital (1, 11, 100)-net over F81, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (15, 36, 250)-net over F81, using
(28, 28+21, 1038)-Net over F27 — Digital
Digital (28, 49, 1038)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2749, 1038, F27, 21) (dual of [1038, 989, 22]-code), using
- 298 step Varšamov–Edel lengthening with (ri) = (3, 1, 4 times 0, 1, 14 times 0, 1, 42 times 0, 1, 90 times 0, 1, 142 times 0) [i] based on linear OA(2741, 732, F27, 21) (dual of [732, 691, 22]-code), using
- construction XX applied to C1 = C([727,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([727,19]) [i] based on
- linear OA(2739, 728, F27, 20) (dual of [728, 689, 21]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,18}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2739, 728, F27, 20) (dual of [728, 689, 21]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2741, 728, F27, 21) (dual of [728, 687, 22]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,19}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(2737, 728, F27, 19) (dual of [728, 691, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 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
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([727,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([727,19]) [i] based on
- 298 step Varšamov–Edel lengthening with (ri) = (3, 1, 4 times 0, 1, 14 times 0, 1, 42 times 0, 1, 90 times 0, 1, 142 times 0) [i] based on linear OA(2741, 732, F27, 21) (dual of [732, 691, 22]-code), using
(28, 28+21, 1292847)-Net in Base 27 — Upper bound on s
There is no (28, 49, 1292848)-net in base 27, because
- 1 times m-reduction [i] would yield (28, 48, 1292848)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 507 529785 063463 962656 177370 339908 333708 383418 746678 980866 401484 333601 > 2748 [i]