Best Known (39−17, 39, s)-Nets in Base 25
(39−17, 39, 192)-Net over F25 — Constructive and digital
Digital (22, 39, 192)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (4, 12, 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, 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 (4, 12, 66)-net over F25, using
(39−17, 39, 760)-Net over F25 — Digital
Digital (22, 39, 760)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2539, 760, F25, 17) (dual of [760, 721, 18]-code), using
- 126 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 10 times 0, 1, 32 times 0, 1, 77 times 0) [i] based on linear OA(2533, 628, F25, 17) (dual of [628, 595, 18]-code), using
- construction XX applied to C1 = C([623,14]), C2 = C([0,15]), C3 = C1 + C2 = C([0,14]), and C∩ = C1 ∩ C2 = C([623,15]) [i] based on
- linear OA(2531, 624, F25, 16) (dual of [624, 593, 17]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,14}, and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2531, 624, F25, 16) (dual of [624, 593, 17]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2533, 624, F25, 17) (dual of [624, 591, 18]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,15}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2529, 624, F25, 15) (dual of [624, 595, 16]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(250, 2, F25, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(250, s, F25, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(250, 2, F25, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([623,14]), C2 = C([0,15]), C3 = C1 + C2 = C([0,14]), and C∩ = C1 ∩ C2 = C([623,15]) [i] based on
- 126 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 10 times 0, 1, 32 times 0, 1, 77 times 0) [i] based on linear OA(2533, 628, F25, 17) (dual of [628, 595, 18]-code), using
(39−17, 39, 685001)-Net in Base 25 — Upper bound on s
There is no (22, 39, 685002)-net in base 25, because
- 1 times m-reduction [i] would yield (22, 38, 685002)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 132349 885224 661552 750976 522997 412095 746744 080492 650625 > 2538 [i]