Best Known (34, 67, s)-Nets in Base 81
(34, 67, 486)-Net over F81 — Constructive and digital
Digital (34, 67, 486)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (2, 18, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-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 (2, 18, 116)-net over F81, using
(34, 67, 2189)-Net over F81 — Digital
Digital (34, 67, 2189)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8167, 2189, F81, 3, 33) (dual of [(2189, 3), 6500, 34]-NRT-code), using
- 811 times duplication [i] 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
- 811 times duplication [i] based on linear OOA(8166, 2189, F81, 3, 33) (dual of [(2189, 3), 6501, 34]-NRT-code), using
(34, 67, 6337951)-Net in Base 81 — Upper bound on s
There is no (34, 67, 6337952)-net in base 81, because
- 1 times m-reduction [i] would yield (34, 66, 6337952)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 912035 561576 044206 730954 162593 974462 088645 539447 202053 306127 443899 440081 966686 729445 168550 167029 118034 131597 251639 382260 162561 > 8166 [i]