Best Known (45−21, 45, s)-Nets in Base 25
(45−21, 45, 192)-Net over F25 — Constructive and digital
Digital (24, 45, 192)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (4, 14, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 4 and N(F) ≥ 66, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- digital (10, 31, 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 (4, 14, 66)-net over F25, using
(45−21, 45, 562)-Net over F25 — Digital
Digital (24, 45, 562)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2545, 562, F25, 21) (dual of [562, 517, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2545, 639, F25, 21) (dual of [639, 594, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- linear OA(2541, 625, F25, 21) (dual of [625, 584, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2531, 625, F25, 16) (dual of [625, 594, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(254, 14, F25, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,25)), using
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- Reed–Solomon code RS(21,25) [i]
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(2545, 639, F25, 21) (dual of [639, 594, 22]-code), using
(45−21, 45, 267111)-Net in Base 25 — Upper bound on s
There is no (24, 45, 267112)-net in base 25, because
- 1 times m-reduction [i] would yield (24, 44, 267112)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 32 311960 850362 191756 140902 407537 219241 952520 828787 083074 816641 > 2544 [i]