Best Known (25, 25+16, s)-Nets in Base 25
(25, 25+16, 252)-Net over F25 — Constructive and digital
Digital (25, 41, 252)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (7, 15, 157)-net over F25, using
- net defined by OOA [i] based on linear OOA(2515, 157, F25, 8, 8) (dual of [(157, 8), 1241, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2515, 628, F25, 8) (dual of [628, 613, 9]-code), using
- construction XX applied to C1 = C([623,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([623,6]) [i] based on
- linear OA(2513, 624, F25, 7) (dual of [624, 611, 8]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(2513, 624, F25, 7) (dual of [624, 611, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(2515, 624, F25, 8) (dual of [624, 609, 9]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(2511, 624, F25, 6) (dual of [624, 613, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [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([623,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([623,6]) [i] based on
- OA 4-folding and stacking [i] based on linear OA(2515, 628, F25, 8) (dual of [628, 613, 9]-code), using
- net defined by OOA [i] based on linear OOA(2515, 157, F25, 8, 8) (dual of [(157, 8), 1241, 9]-NRT-code), using
- digital (10, 26, 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 (7, 15, 157)-net over F25, using
(25, 25+16, 1780)-Net over F25 — Digital
Digital (25, 41, 1780)-net over F25, using
(25, 25+16, 2290449)-Net in Base 25 — Upper bound on s
There is no (25, 41, 2290450)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 2067 958172 738626 884434 414567 660987 369421 896425 448476 344961 > 2541 [i]