Best Known (64−34, 64, s)-Nets in Base 27
(64−34, 64, 158)-Net over F27 — Constructive and digital
Digital (30, 64, 158)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (6, 23, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 6 and N(F) ≥ 76, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- digital (7, 41, 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 (6, 23, 76)-net over F27, using
(64−34, 64, 224)-Net in Base 27 — Constructive
(30, 64, 224)-net in base 27, using
- 4 times m-reduction [i] based on (30, 68, 224)-net in base 27, using
- base change [i] based on digital (13, 51, 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, 51, 224)-net over F81, using
(64−34, 64, 333)-Net over F27 — Digital
Digital (30, 64, 333)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2764, 333, F27, 2, 34) (dual of [(333, 2), 602, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2764, 366, F27, 2, 34) (dual of [(366, 2), 668, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2764, 732, F27, 34) (dual of [732, 668, 35]-code), using
- construction XX applied to C1 = C([727,31]), C2 = C([0,32]), C3 = C1 + C2 = C([0,31]), and C∩ = C1 ∩ C2 = C([727,32]) [i] based on
- linear OA(2762, 728, F27, 33) (dual of [728, 666, 34]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,31}, and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(2762, 728, F27, 33) (dual of [728, 666, 34]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,32], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(2764, 728, F27, 34) (dual of [728, 664, 35]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,32}, and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2760, 728, F27, 32) (dual of [728, 668, 33]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,31], and designed minimum distance d ≥ |I|+1 = 33 [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([727,31]), C2 = C([0,32]), C3 = C1 + C2 = C([0,31]), and C∩ = C1 ∩ C2 = C([727,32]) [i] based on
- OOA 2-folding [i] based on linear OA(2764, 732, F27, 34) (dual of [732, 668, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(2764, 366, F27, 2, 34) (dual of [(366, 2), 668, 35]-NRT-code), using
(64−34, 64, 67542)-Net in Base 27 — Upper bound on s
There is no (30, 64, 67543)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 40 484626 531700 852838 956818 962282 460285 066430 317275 689519 435412 088805 861864 222375 559021 584439 > 2764 [i]