Best Known (33−19, 33, s)-Nets in Base 81
(33−19, 33, 250)-Net over F81 — Constructive and digital
Digital (14, 33, 250)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- digital (4, 23, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- digital (1, 10, 100)-net over F81, using
(33−19, 33, 332)-Net over F81 — Digital
Digital (14, 33, 332)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8133, 332, F81, 19) (dual of [332, 299, 20]-code), using
- construction XX applied to C1 = C([33,50]), C2 = C([32,49]), C3 = C1 + C2 = C([33,49]), and C∩ = C1 ∩ C2 = C([32,50]) [i] based on
- linear OA(8131, 328, F81, 18) (dual of [328, 297, 19]-code), using the BCH-code C(I) with length 328 | 812−1, defining interval I = {33,34,…,50}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(8131, 328, F81, 18) (dual of [328, 297, 19]-code), using the BCH-code C(I) with length 328 | 812−1, defining interval I = {32,33,…,49}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(8133, 328, F81, 19) (dual of [328, 295, 20]-code), using the BCH-code C(I) with length 328 | 812−1, defining interval I = {32,33,…,50}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(8129, 328, F81, 17) (dual of [328, 299, 18]-code), using the BCH-code C(I) with length 328 | 812−1, defining interval I = {33,34,…,49}, and designed minimum distance d ≥ |I|+1 = 18 [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([33,50]), C2 = C([32,49]), C3 = C1 + C2 = C([33,49]), and C∩ = C1 ∩ C2 = C([32,50]) [i] based on
(33−19, 33, 316504)-Net in Base 81 — Upper bound on s
There is no (14, 33, 316505)-net in base 81, because
- 1 times m-reduction [i] would yield (14, 32, 316505)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 11 790244 551198 529915 995319 045334 121517 299951 414340 071413 419601 > 8132 [i]