Best Known (66−31, 66, s)-Nets in Base 25
(66−31, 66, 252)-Net over F25 — Constructive and digital
Digital (35, 66, 252)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (10, 25, 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, 41, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25 (see above)
- digital (10, 25, 126)-net over F25, using
(66−31, 66, 648)-Net over F25 — Digital
Digital (35, 66, 648)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2566, 648, F25, 31) (dual of [648, 582, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(21) [i] based on
- 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(2543, 625, F25, 22) (dual of [625, 582, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(258, 23, F25, 8) (dual of [23, 15, 9]-code or 23-arc in PG(7,25)), using
- discarding factors / shortening the dual code based on linear OA(258, 25, F25, 8) (dual of [25, 17, 9]-code or 25-arc in PG(7,25)), using
- Reed–Solomon code RS(17,25) [i]
- discarding factors / shortening the dual code based on linear OA(258, 25, F25, 8) (dual of [25, 17, 9]-code or 25-arc in PG(7,25)), using
- construction X applied to Ce(30) ⊂ Ce(21) [i] based on
(66−31, 66, 305692)-Net in Base 25 — Upper bound on s
There is no (35, 66, 305693)-net in base 25, because
- 1 times m-reduction [i] would yield (35, 65, 305693)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 7 347065 524658 096257 008223 804568 371574 386665 115161 929404 490220 435888 279490 672358 986904 014729 > 2565 [i]