Best Known (72−38, 72, s)-Nets in Base 27
(72−38, 72, 166)-Net over F27 — Constructive and digital
Digital (34, 72, 166)-net over F27, using
- (u, u+v)-construction [i] based on
- 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 (8, 46, 84)-net over F27, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 8 and N(F) ≥ 84, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- digital (7, 26, 82)-net over F27, using
(72−38, 72, 366)-Net over F27 — Digital
Digital (34, 72, 366)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2772, 366, F27, 2, 38) (dual of [(366, 2), 660, 39]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2772, 732, F27, 38) (dual of [732, 660, 39]-code), using
- construction XX applied to C1 = C([727,35]), C2 = C([0,36]), C3 = C1 + C2 = C([0,35]), and C∩ = C1 ∩ C2 = C([727,36]) [i] based on
- linear OA(2770, 728, F27, 37) (dual of [728, 658, 38]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,35}, and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(2770, 728, F27, 37) (dual of [728, 658, 38]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,36], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(2772, 728, F27, 38) (dual of [728, 656, 39]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,36}, and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(2768, 728, F27, 36) (dual of [728, 660, 37]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,35], and designed minimum distance d ≥ |I|+1 = 37 [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([727,35]), C2 = C([0,36]), C3 = C1 + C2 = C([0,35]), and C∩ = C1 ∩ C2 = C([727,36]) [i] based on
- OOA 2-folding [i] based on linear OA(2772, 732, F27, 38) (dual of [732, 660, 39]-code), using
(72−38, 72, 370)-Net in Base 27 — Constructive
(34, 72, 370)-net in base 27, using
- base change [i] based on digital (16, 54, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
(72−38, 72, 80966)-Net in Base 27 — Upper bound on s
There is no (34, 72, 80967)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 11 435193 250255 026870 941236 847489 180552 705548 081820 036079 374233 974025 920426 727997 213890 716399 702790 404979 > 2772 [i]