Best Known (32, 32+25, s)-Nets in Base 25
(32, 32+25, 252)-Net over F25 — Constructive and digital
Digital (32, 57, 252)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (10, 22, 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, 35, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25 (see above)
- digital (10, 22, 126)-net over F25, using
(32, 32+25, 868)-Net over F25 — Digital
Digital (32, 57, 868)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2557, 868, F25, 25) (dual of [868, 811, 26]-code), using
- 235 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 6 times 0, 1, 18 times 0, 1, 38 times 0, 1, 68 times 0, 1, 97 times 0) [i] based on linear OA(2548, 624, F25, 25) (dual of [624, 576, 26]-code), using
- 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]
- 235 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 6 times 0, 1, 18 times 0, 1, 38 times 0, 1, 68 times 0, 1, 97 times 0) [i] based on linear OA(2548, 624, F25, 25) (dual of [624, 576, 26]-code), using
(32, 32+25, 735981)-Net in Base 25 — Upper bound on s
There is no (32, 57, 735982)-net in base 25, because
- 1 times m-reduction [i] would yield (32, 56, 735982)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 1 925957 667312 164479 849939 308689 196075 005452 922809 694427 880532 572200 498925 276225 > 2556 [i]