Best Known (42, 42+39, s)-Nets in Base 25
(42, 42+39, 288)-Net over F25 — Constructive and digital
Digital (42, 81, 288)-net over F25, using
- t-expansion [i] based on digital (41, 81, 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
(42, 42+39, 626)-Net over F25 — Digital
Digital (42, 81, 626)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2581, 626, F25, 39) (dual of [626, 545, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(2581, 648, F25, 39) (dual of [648, 567, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(30) [i] based on
- linear OA(2574, 625, F25, 39) (dual of [625, 551, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(2558, 625, F25, 31) (dual of [625, 567, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(257, 23, F25, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,25)), using
- discarding factors / shortening the dual code based on linear OA(257, 25, F25, 7) (dual of [25, 18, 8]-code or 25-arc in PG(6,25)), using
- Reed–Solomon code RS(18,25) [i]
- discarding factors / shortening the dual code based on linear OA(257, 25, F25, 7) (dual of [25, 18, 8]-code or 25-arc in PG(6,25)), using
- construction X applied to Ce(38) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(2581, 648, F25, 39) (dual of [648, 567, 40]-code), using
(42, 42+39, 254126)-Net in Base 25 — Upper bound on s
There is no (42, 81, 254127)-net in base 25, because
- 1 times m-reduction [i] would yield (42, 80, 254127)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 6842 302704 216487 856023 125845 396838 352236 166714 192749 965059 963125 457528 576092 555037 471998 202692 354480 890131 293625 > 2580 [i]