Best Known (21, 21+22, s)-Nets in Base 27
(21, 21+22, 140)-Net over F27 — Constructive and digital
Digital (21, 43, 140)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 15, 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 (6, 28, 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 (4, 15, 64)-net over F27, using
(21, 21+22, 172)-Net in Base 27 — Constructive
(21, 43, 172)-net in base 27, using
- 13 times m-reduction [i] based on (21, 56, 172)-net in base 27, using
- base change [i] based on digital (7, 42, 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, 42, 172)-net over F81, using
(21, 21+22, 365)-Net over F27 — Digital
Digital (21, 43, 365)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2743, 365, F27, 2, 22) (dual of [(365, 2), 687, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2743, 366, F27, 2, 22) (dual of [(366, 2), 689, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2743, 732, F27, 22) (dual of [732, 689, 23]-code), using
- construction XX applied to C1 = C([727,19]), C2 = C([0,20]), C3 = C1 + C2 = C([0,19]), and C∩ = C1 ∩ C2 = C([727,20]) [i] based on
- 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(2741, 728, F27, 21) (dual of [728, 687, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(2743, 728, F27, 22) (dual of [728, 685, 23]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,20}, and designed minimum distance d ≥ |I|+1 = 23 [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(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([727,19]), C2 = C([0,20]), C3 = C1 + C2 = C([0,19]), and C∩ = C1 ∩ C2 = C([727,20]) [i] based on
- OOA 2-folding [i] based on linear OA(2743, 732, F27, 22) (dual of [732, 689, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(2743, 366, F27, 2, 22) (dual of [(366, 2), 689, 23]-NRT-code), using
(21, 21+22, 74360)-Net in Base 27 — Upper bound on s
There is no (21, 43, 74361)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 35 375038 944543 177013 165747 764466 966516 705199 208725 394965 212131 > 2743 [i]