Best Known (20, 29, s)-Nets in Base 9
(20, 29, 364)-Net over F9 — Constructive and digital
Digital (20, 29, 364)-net over F9, using
- 91 times duplication [i] based on digital (19, 28, 364)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (4, 8, 164)-net over F9, using
- trace code for nets [i] based on digital (0, 4, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 4, 82)-net over F81, using
- digital (11, 20, 200)-net over F9, using
- trace code for nets [i] based on digital (1, 10, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 10, 100)-net over F81, using
- digital (4, 8, 164)-net over F9, using
- (u, u+v)-construction [i] based on
(20, 29, 400)-Net in Base 9 — Constructive
(20, 29, 400)-net in base 9, using
- (u, u+v)-construction [i] based on
- (5, 9, 351)-net in base 9, using
- base change [i] based on digital (2, 6, 351)-net over F27, using
- net defined by OOA [i] based on linear OOA(276, 351, F27, 4, 4) (dual of [(351, 4), 1398, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(276, 702, F27, 4) (dual of [702, 696, 5]-code), using
- 1 times truncation [i] based on linear OA(277, 703, F27, 5) (dual of [703, 696, 6]-code), using
- OA 2-folding and stacking [i] based on linear OA(276, 702, F27, 4) (dual of [702, 696, 5]-code), using
- net defined by OOA [i] based on linear OOA(276, 351, F27, 4, 4) (dual of [(351, 4), 1398, 5]-NRT-code), using
- base change [i] based on digital (2, 6, 351)-net over F27, using
- digital (11, 20, 200)-net over F9, using
- trace code for nets [i] based on digital (1, 10, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 10, 100)-net over F81, using
- (5, 9, 351)-net in base 9, using
(20, 29, 1360)-Net over F9 — Digital
Digital (20, 29, 1360)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(929, 1360, F9, 9) (dual of [1360, 1331, 10]-code), using
- 622 step Varšamov–Edel lengthening with (ri) = (1, 13 times 0, 1, 80 times 0, 1, 208 times 0, 1, 317 times 0) [i] based on linear OA(925, 734, F9, 9) (dual of [734, 709, 10]-code), using
- construction XX applied to C1 = C([727,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([727,7]) [i] based on
- linear OA(922, 728, F9, 8) (dual of [728, 706, 9]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(922, 728, F9, 8) (dual of [728, 706, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(925, 728, F9, 9) (dual of [728, 703, 10]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,7}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(919, 728, F9, 7) (dual of [728, 709, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(90, 3, F9, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(90, 3, F9, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([727,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([727,7]) [i] based on
- 622 step Varšamov–Edel lengthening with (ri) = (1, 13 times 0, 1, 80 times 0, 1, 208 times 0, 1, 317 times 0) [i] based on linear OA(925, 734, F9, 9) (dual of [734, 709, 10]-code), using
(20, 29, 1323304)-Net in Base 9 — Upper bound on s
There is no (20, 29, 1323305)-net in base 9, because
- 1 times m-reduction [i] would yield (20, 28, 1323305)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 523 348694 987479 983638 415521 > 928 [i]