Best Known (23, 40, s)-Nets in Base 25
(23, 40, 198)-Net over F25 — Constructive and digital
Digital (23, 40, 198)-net over F25, using
- generalized (u, u+v)-construction [i] based on
- digital (2, 7, 66)-net over F25, using
- s-reduction based on digital (2, 7, 300)-net over F25, using
- net defined by OOA [i] based on linear OOA(257, 300, F25, 5, 5) (dual of [(300, 5), 1493, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(257, 601, F25, 5) (dual of [601, 594, 6]-code), using
- net defined by OOA [i] based on linear OOA(257, 300, F25, 5, 5) (dual of [(300, 5), 1493, 6]-NRT-code), using
- s-reduction based on digital (2, 7, 300)-net over F25, using
- digital (4, 12, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 4 and N(F) ≥ 66, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- digital (4, 21, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25 (see above)
- digital (2, 7, 66)-net over F25, using
(23, 40, 899)-Net over F25 — Digital
Digital (23, 40, 899)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2540, 899, F25, 17) (dual of [899, 859, 18]-code), using
- 264 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) [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
- 264 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) [i] based on linear OA(2533, 628, F25, 17) (dual of [628, 595, 18]-code), using
(23, 40, 1024317)-Net in Base 25 — Upper bound on s
There is no (23, 40, 1024318)-net in base 25, because
- 1 times m-reduction [i] would yield (23, 39, 1024318)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 3 308726 336929 717476 621595 143540 898274 504040 737084 083585 > 2539 [i]