Best Known (41−17, 41, s)-Nets in Base 25
(41−17, 41, 208)-Net over F25 — Constructive and digital
Digital (24, 41, 208)-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 (9, 26, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- digital (7, 15, 157)-net over F25, using
(41−17, 41, 1092)-Net over F25 — Digital
Digital (24, 41, 1092)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2541, 1092, F25, 17) (dual of [1092, 1051, 18]-code), using
- 456 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 10 times 0, 1, 32 times 0, 1, 77 times 0, 1, 137 times 0, 1, 191 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
- 456 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 10 times 0, 1, 32 times 0, 1, 77 times 0, 1, 137 times 0, 1, 191 times 0) [i] based on linear OA(2533, 628, F25, 17) (dual of [628, 595, 18]-code), using
(41−17, 41, 1531714)-Net in Base 25 — Upper bound on s
There is no (24, 41, 1531715)-net in base 25, because
- 1 times m-reduction [i] would yield (24, 40, 1531715)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 82 718343 453742 995723 817063 947531 294941 752786 922929 998913 > 2540 [i]