Best Known (71−34, 71, s)-Nets in Base 25
(71−34, 71, 252)-Net over F25 — Constructive and digital
Digital (37, 71, 252)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (10, 27, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- digital (10, 44, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25 (see above)
- digital (10, 27, 126)-net over F25, using
(71−34, 71, 594)-Net over F25 — Digital
Digital (37, 71, 594)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2571, 594, F25, 34) (dual of [594, 523, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(2571, 647, F25, 34) (dual of [647, 576, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(25) [i] based on
- linear OA(2564, 625, F25, 34) (dual of [625, 561, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(2549, 625, F25, 26) (dual of [625, 576, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(257, 22, F25, 7) (dual of [22, 15, 8]-code or 22-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(33) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(2571, 647, F25, 34) (dual of [647, 576, 35]-code), using
(71−34, 71, 206144)-Net in Base 25 — Upper bound on s
There is no (37, 71, 206145)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 1793 792336 696007 357274 308947 076286 656768 376544 374304 847771 649636 464221 952038 026301 982408 483319 480729 > 2571 [i]