Best Known (82−40, 82, s)-Nets in Base 25
(82−40, 82, 288)-Net over F25 — Constructive and digital
Digital (42, 82, 288)-net over F25, using
- t-expansion [i] based on digital (41, 82, 288)-net over F25, using
- net from sequence [i] based on digital (41, 287)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 41 and N(F) ≥ 288, using
- net from sequence [i] based on digital (41, 287)-sequence over F25, using
(82−40, 82, 580)-Net over F25 — Digital
Digital (42, 82, 580)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2582, 580, F25, 40) (dual of [580, 498, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(2582, 645, F25, 40) (dual of [645, 563, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(32) [i] based on
- linear OA(2576, 625, F25, 40) (dual of [625, 549, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(2562, 625, F25, 33) (dual of [625, 563, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(256, 20, F25, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,25)), using
- discarding factors / shortening the dual code based on linear OA(256, 25, F25, 6) (dual of [25, 19, 7]-code or 25-arc in PG(5,25)), using
- Reed–Solomon code RS(19,25) [i]
- discarding factors / shortening the dual code based on linear OA(256, 25, F25, 6) (dual of [25, 19, 7]-code or 25-arc in PG(5,25)), using
- construction X applied to Ce(39) ⊂ Ce(32) [i] based on
- discarding factors / shortening the dual code based on linear OA(2582, 645, F25, 40) (dual of [645, 563, 41]-code), using
(82−40, 82, 186476)-Net in Base 25 — Upper bound on s
There is no (42, 82, 186477)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 4 276485 340429 283485 247306 895155 012248 793298 406945 946910 045609 647363 468285 259324 273130 674857 848799 426676 349382 034145 > 2582 [i]