Best Known (51−27, 51, s)-Nets in Base 81
(51−27, 51, 370)-Net over F81 — Constructive and digital
Digital (24, 51, 370)-net over F81, using
- t-expansion [i] based on digital (16, 51, 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
(51−27, 51, 824)-Net over F81 — Digital
Digital (24, 51, 824)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8151, 824, F81, 27) (dual of [824, 773, 28]-code), using
- construction XX applied to C1 = C([27,52]), C2 = C([26,51]), C3 = C1 + C2 = C([27,51]), and C∩ = C1 ∩ C2 = C([26,52]) [i] based on
- linear OA(8149, 820, F81, 26) (dual of [820, 771, 27]-code), using the BCH-code C(I) with length 820 | 812−1, defining interval I = {27,28,…,52}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(8149, 820, F81, 26) (dual of [820, 771, 27]-code), using the BCH-code C(I) with length 820 | 812−1, defining interval I = {26,27,…,51}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(8151, 820, F81, 27) (dual of [820, 769, 28]-code), using the BCH-code C(I) with length 820 | 812−1, defining interval I = {26,27,…,52}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(8147, 820, F81, 25) (dual of [820, 773, 26]-code), using the BCH-code C(I) with length 820 | 812−1, defining interval I = {27,28,…,51}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([27,52]), C2 = C([26,51]), C3 = C1 + C2 = C([27,51]), and C∩ = C1 ∩ C2 = C([26,52]) [i] based on
(51−27, 51, 1551105)-Net in Base 81 — Upper bound on s
There is no (24, 51, 1551106)-net in base 81, because
- 1 times m-reduction [i] would yield (24, 50, 1551106)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 265614 118830 812035 866906 029230 407157 413997 898757 514109 078416 544437 308218 661380 012346 958621 905441 > 8150 [i]