Best Known (26, 26+19, s)-Nets in Base 25
(26, 26+19, 208)-Net over F25 — Constructive and digital
Digital (26, 45, 208)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (8, 17, 156)-net over F25, using
- net defined by OOA [i] based on linear OOA(2517, 156, F25, 9, 9) (dual of [(156, 9), 1387, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2517, 625, F25, 9) (dual of [625, 608, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(2517, 625, F25, 9) (dual of [625, 608, 10]-code), using
- net defined by OOA [i] based on linear OOA(2517, 156, F25, 9, 9) (dual of [(156, 9), 1387, 10]-NRT-code), using
- digital (9, 28, 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 (8, 17, 156)-net over F25, using
(26, 26+19, 993)-Net over F25 — Digital
Digital (26, 45, 993)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2545, 993, F25, 19) (dual of [993, 948, 20]-code), using
- 357 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 7 times 0, 1, 23 times 0, 1, 57 times 0, 1, 109 times 0, 1, 153 times 0) [i] based on linear OA(2537, 628, F25, 19) (dual of [628, 591, 20]-code), using
- construction XX applied to C1 = C([623,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([623,17]) [i] based on
- linear OA(2535, 624, F25, 18) (dual of [624, 589, 19]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,16}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2535, 624, F25, 18) (dual of [624, 589, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2537, 624, F25, 19) (dual of [624, 587, 20]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,17}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(2533, 624, F25, 17) (dual of [624, 591, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [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([623,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([623,17]) [i] based on
- 357 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 7 times 0, 1, 23 times 0, 1, 57 times 0, 1, 109 times 0, 1, 153 times 0) [i] based on linear OA(2537, 628, F25, 19) (dual of [628, 591, 20]-code), using
(26, 26+19, 1180081)-Net in Base 25 — Upper bound on s
There is no (26, 45, 1180082)-net in base 25, because
- 1 times m-reduction [i] would yield (26, 44, 1180082)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 32 311885 982582 290353 499762 620927 843396 221663 876289 450220 575025 > 2544 [i]