Best Known (45−23, 45, s)-Nets in Base 25
(45−23, 45, 153)-Net over F25 — Constructive and digital
Digital (22, 45, 153)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (1, 12, 27)-net over F25, using
- net from sequence [i] based on digital (1, 26)-sequence over F25, using
- digital (10, 33, 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 (1, 12, 27)-net over F25, using
(45−23, 45, 314)-Net over F25 — Digital
Digital (22, 45, 314)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2545, 314, F25, 2, 23) (dual of [(314, 2), 583, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2545, 628, F25, 23) (dual of [628, 583, 24]-code), using
- construction XX applied to C1 = C([623,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([623,21]) [i] based on
- linear OA(2543, 624, F25, 22) (dual of [624, 581, 23]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,20}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2543, 624, F25, 22) (dual of [624, 581, 23]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2545, 624, F25, 23) (dual of [624, 579, 24]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,21}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(2541, 624, F25, 21) (dual of [624, 583, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [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([623,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([623,21]) [i] based on
- OOA 2-folding [i] based on linear OA(2545, 628, F25, 23) (dual of [628, 583, 24]-code), using
(45−23, 45, 79897)-Net in Base 25 — Upper bound on s
There is no (22, 45, 79898)-net in base 25, because
- 1 times m-reduction [i] would yield (22, 44, 79898)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 32 313892 643429 020868 083846 106653 893200 955091 646460 006518 034193 > 2544 [i]