Best Known (21, 21+17, s)-Nets in Base 25
(21, 21+17, 178)-Net over F25 — Constructive and digital
Digital (21, 38, 178)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (3, 11, 52)-net over F25, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-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 (3, 11, 52)-net over F25, using
(21, 21+17, 681)-Net over F25 — Digital
Digital (21, 38, 681)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2538, 681, F25, 17) (dual of [681, 643, 18]-code), using
- 48 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 10 times 0, 1, 32 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
- 48 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 10 times 0, 1, 32 times 0) [i] based on linear OA(2533, 628, F25, 17) (dual of [628, 595, 18]-code), using
(21, 21+17, 458086)-Net in Base 25 — Upper bound on s
There is no (21, 38, 458087)-net in base 25, because
- 1 times m-reduction [i] would yield (21, 37, 458087)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 5293 979084 991334 165282 128313 911362 461749 967183 520065 > 2537 [i]