Best Known (30−12, 30, s)-Nets in Base 27
(30−12, 30, 196)-Net over F27 — Constructive and digital
Digital (18, 30, 196)-net over F27, using
- 1 times m-reduction [i] based on digital (18, 31, 196)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 28)-net over F27, using
- s-reduction based on digital (0, 1, s)-net over F27 with arbitrarily large s, using
- digital (0, 2, 28)-net over F27, using
- digital (0, 2, 28)-net over F27 (see above)
- 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, 4, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 6, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 13, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 1, 28)-net over F27, using
- generalized (u, u+v)-construction [i] based on
(30−12, 30, 232)-Net in Base 27 — Constructive
(18, 30, 232)-net in base 27, using
- (u, u+v)-construction [i] based on
- digital (5, 11, 244)-net over F27, using
- net defined by OOA [i] based on linear OOA(2711, 244, F27, 6, 6) (dual of [(244, 6), 1453, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2711, 732, F27, 6) (dual of [732, 721, 7]-code), using
- construction XX applied to C1 = C([727,3]), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([727,4]) [i] based on
- linear OA(279, 728, F27, 5) (dual of [728, 719, 6]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,1,2,3}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(279, 728, F27, 5) (dual of [728, 719, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2711, 728, F27, 6) (dual of [728, 717, 7]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,4}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(277, 728, F27, 4) (dual of [728, 721, 5]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [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([727,3]), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([727,4]) [i] based on
- OA 3-folding and stacking [i] based on linear OA(2711, 732, F27, 6) (dual of [732, 721, 7]-code), using
- net defined by OOA [i] based on linear OOA(2711, 244, F27, 6, 6) (dual of [(244, 6), 1453, 7]-NRT-code), using
- (7, 19, 116)-net in base 27, using
- 1 times m-reduction [i] based on (7, 20, 116)-net in base 27, using
- base change [i] based on digital (2, 15, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- base change [i] based on digital (2, 15, 116)-net over F81, using
- 1 times m-reduction [i] based on (7, 20, 116)-net in base 27, using
- digital (5, 11, 244)-net over F27, using
(30−12, 30, 1519)-Net over F27 — Digital
Digital (18, 30, 1519)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2730, 1519, F27, 12) (dual of [1519, 1489, 13]-code), using
- 780 step Varšamov–Edel lengthening with (ri) = (3, 5 times 0, 1, 27 times 0, 1, 108 times 0, 1, 251 times 0, 1, 384 times 0) [i] based on linear OA(2723, 732, F27, 12) (dual of [732, 709, 13]-code), using
- construction XX applied to C1 = C([727,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([727,10]) [i] based on
- linear OA(2721, 728, F27, 11) (dual of [728, 707, 12]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,9}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2721, 728, F27, 11) (dual of [728, 707, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2723, 728, F27, 12) (dual of [728, 705, 13]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,10}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2719, 728, F27, 10) (dual of [728, 709, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [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([727,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([727,10]) [i] based on
- 780 step Varšamov–Edel lengthening with (ri) = (3, 5 times 0, 1, 27 times 0, 1, 108 times 0, 1, 251 times 0, 1, 384 times 0) [i] based on linear OA(2723, 732, F27, 12) (dual of [732, 709, 13]-code), using
(30−12, 30, 1652216)-Net in Base 27 — Upper bound on s
There is no (18, 30, 1652217)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 8 727993 548123 290513 454118 989188 125304 237305 > 2730 [i]