Best Known (18, 37, s)-Nets in Base 27
(18, 37, 132)-Net over F27 — Constructive and digital
Digital (18, 37, 132)-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 (5, 24, 68)-net over F27, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 5 and N(F) ≥ 68, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- digital (4, 13, 64)-net over F27, using
(18, 37, 172)-Net in Base 27 — Constructive
(18, 37, 172)-net in base 27, using
- 7 times m-reduction [i] based on (18, 44, 172)-net in base 27, using
- base change [i] based on digital (7, 33, 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, 33, 172)-net over F81, using
(18, 37, 346)-Net over F27 — Digital
Digital (18, 37, 346)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2737, 346, F27, 2, 19) (dual of [(346, 2), 655, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2737, 366, F27, 2, 19) (dual of [(366, 2), 695, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2737, 732, F27, 19) (dual of [732, 695, 20]-code), using
- construction XX applied to C1 = C([727,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([727,17]) [i] based on
- linear OA(2735, 728, F27, 18) (dual of [728, 693, 19]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,16}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2735, 728, F27, 18) (dual of [728, 693, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2737, 728, F27, 19) (dual of [728, 691, 20]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,17}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(2733, 728, F27, 17) (dual of [728, 695, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [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([727,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([727,17]) [i] based on
- OOA 2-folding [i] based on linear OA(2737, 732, F27, 19) (dual of [732, 695, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(2737, 366, F27, 2, 19) (dual of [(366, 2), 695, 20]-NRT-code), using
(18, 37, 84763)-Net in Base 27 — Upper bound on s
There is no (18, 37, 84764)-net in base 27, because
- 1 times m-reduction [i] would yield (18, 36, 84764)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 3381 430666 004458 789436 202864 848745 059303 993191 475033 > 2736 [i]