Best Known (66−33, 66, s)-Nets in Base 81
(66−33, 66, 470)-Net over F81 — Constructive and digital
Digital (33, 66, 470)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 17, 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 (16, 49, 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
- digital (1, 17, 100)-net over F81, using
(66−33, 66, 2028)-Net over F81 — Digital
Digital (33, 66, 2028)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8166, 2028, F81, 3, 33) (dual of [(2028, 3), 6018, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8166, 2189, F81, 3, 33) (dual of [(2189, 3), 6501, 34]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8166, 6567, F81, 33) (dual of [6567, 6501, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,15]) [i] based on
- linear OA(8165, 6562, F81, 33) (dual of [6562, 6497, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(8161, 6562, F81, 31) (dual of [6562, 6501, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,16]) ⊂ C([0,15]) [i] based on
- OOA 3-folding [i] based on linear OA(8166, 6567, F81, 33) (dual of [6567, 6501, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(8166, 2189, F81, 3, 33) (dual of [(2189, 3), 6501, 34]-NRT-code), using
(66−33, 66, 4815799)-Net in Base 81 — Upper bound on s
There is no (33, 66, 4815800)-net in base 81, because
- 1 times m-reduction [i] would yield (33, 65, 4815800)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 11259 688778 571123 371865 711881 607712 705079 699061 821633 810638 534481 598400 582864 884049 059961 371416 772213 288418 943343 627207 424001 > 8165 [i]