Best Known (20, 20+21, s)-Nets in Base 25
(20, 20+21, 152)-Net over F25 — Constructive and digital
Digital (20, 41, 152)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 26)-net over F25, using
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using
- the rational function field F25(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 25)-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 (0, 10, 26)-net over F25, using
(20, 20+21, 314)-Net over F25 — Digital
Digital (20, 41, 314)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2541, 314, F25, 2, 21) (dual of [(314, 2), 587, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2541, 628, F25, 21) (dual of [628, 587, 22]-code), using
- construction XX applied to C1 = C([623,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([623,19]) [i] based on
- linear OA(2539, 624, F25, 20) (dual of [624, 585, 21]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,18}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2539, 624, F25, 20) (dual of [624, 585, 21]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2541, 624, F25, 21) (dual of [624, 583, 22]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,19}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(2537, 624, F25, 19) (dual of [624, 587, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [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([623,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([623,19]) [i] based on
- OOA 2-folding [i] based on linear OA(2541, 628, F25, 21) (dual of [628, 587, 22]-code), using
(20, 20+21, 73704)-Net in Base 25 — Upper bound on s
There is no (20, 41, 73705)-net in base 25, because
- 1 times m-reduction [i] would yield (20, 40, 73705)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 82 719629 666572 211380 743367 817758 767241 928623 558842 608305 > 2540 [i]