Best Known (32, 32+31, s)-Nets in Base 81
(32, 32+31, 470)-Net over F81 — Constructive and digital
Digital (32, 63, 470)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 16, 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, 47, 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, 16, 100)-net over F81, using
(32, 32+31, 2189)-Net over F81 — Digital
Digital (32, 63, 2189)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8163, 2189, F81, 3, 31) (dual of [(2189, 3), 6504, 32]-NRT-code), using
- 811 times duplication [i] based on linear OOA(8162, 2189, F81, 3, 31) (dual of [(2189, 3), 6505, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8162, 6567, F81, 31) (dual of [6567, 6505, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,14]) [i] based on
- 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(8157, 6562, F81, 29) (dual of [6562, 6505, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (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([0,15]) ⊂ C([0,14]) [i] based on
- OOA 3-folding [i] based on linear OA(8162, 6567, F81, 31) (dual of [6567, 6505, 32]-code), using
- 811 times duplication [i] based on linear OOA(8162, 2189, F81, 3, 31) (dual of [(2189, 3), 6505, 32]-NRT-code), using
(32, 32+31, 6209846)-Net in Base 81 — Upper bound on s
There is no (32, 63, 6209847)-net in base 81, because
- 1 times m-reduction [i] would yield (32, 62, 6209847)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 21187 086238 073736 002641 214753 378130 772326 443483 424809 753370 734794 077825 046526 511170 307355 611924 692884 399856 242136 872401 > 8162 [i]