Best Known (12, 12+9, s)-Nets in Base 25
(12, 12+9, 182)-Net over F25 — Constructive and digital
Digital (12, 21, 182)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 26)-net over F25, using
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using
- the rational function field F25(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- 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 (0, 4, 26)-net over F25, using
(12, 12+9, 797)-Net over F25 — Digital
Digital (12, 21, 797)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2521, 797, F25, 9) (dual of [797, 776, 10]-code), using
- 165 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 32 times 0, 1, 125 times 0) [i] based on linear OA(2517, 628, F25, 9) (dual of [628, 611, 10]-code), using
- construction XX applied to C1 = C([623,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([623,7]) [i] based on
- 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(2515, 624, F25, 8) (dual of [624, 609, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(2517, 624, F25, 9) (dual of [624, 607, 10]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,7}, and designed minimum distance d ≥ |I|+1 = 10 [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(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([623,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([623,7]) [i] based on
- 165 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 32 times 0, 1, 125 times 0) [i] based on linear OA(2517, 628, F25, 9) (dual of [628, 611, 10]-code), using
(12, 12+9, 900618)-Net in Base 25 — Upper bound on s
There is no (12, 21, 900619)-net in base 25, because
- 1 times m-reduction [i] would yield (12, 20, 900619)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 9094 971853 142145 555476 061985 > 2520 [i]