Best Known (24−7, 24, s)-Nets in Base 9
(24−7, 24, 738)-Net over F9 — Constructive and digital
Digital (17, 24, 738)-net over F9, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 82)-net over F9, using
- s-reduction based on digital (0, 0, s)-net over F9 with arbitrarily large s, using
- digital (0, 0, 82)-net over F9 (see above)
- digital (0, 1, 82)-net over F9, using
- s-reduction based on digital (0, 1, s)-net over F9 with arbitrarily large s, using
- digital (0, 1, 82)-net over F9 (see above)
- digital (0, 1, 82)-net over F9 (see above)
- digital (0, 1, 82)-net over F9 (see above)
- digital (1, 3, 82)-net over F9, using
- s-reduction based on digital (1, 3, 91)-net over F9, using
- digital (1, 4, 82)-net over F9, using
- net defined by OOA [i] based on linear OOA(94, 82, F9, 3, 3) (dual of [(82, 3), 242, 4]-NRT-code), using
- digital (6, 13, 82)-net over F9, using
- base reduction for projective spaces (embedding PG(6,81) in PG(12,9)) for nets [i] based on digital (0, 7, 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
- base reduction for projective spaces (embedding PG(6,81) in PG(12,9)) for nets [i] based on digital (0, 7, 82)-net over F81, using
- digital (0, 0, 82)-net over F9, using
(24−7, 24, 756)-Net in Base 9 — Constructive
(17, 24, 756)-net in base 9, using
- base change [i] based on digital (9, 16, 756)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 28)-net over F27, using
- s-reduction based on digital (0, 0, s)-net over F27 with arbitrarily large s, using
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 1, 28)-net over F27, using
- s-reduction based on digital (0, 1, s)-net over F27 with arbitrarily large s, using
- digital (0, 1, 28)-net over F27 (see above)
- digital (0, 1, 28)-net over F27 (see above)
- digital (0, 1, 28)-net over F27 (see above)
- digital (0, 2, 28)-net over F27, using
- digital (0, 3, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 0 and N(F) ≥ 28, using
- the rational function field F27(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- digital (0, 7, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 0, 28)-net over F27, using
- generalized (u, u+v)-construction [i] based on
(24−7, 24, 2459)-Net over F9 — Digital
Digital (17, 24, 2459)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(924, 2459, F9, 7) (dual of [2459, 2435, 8]-code), using
- 1720 step Varšamov–Edel lengthening with (ri) = (1, 20 times 0, 1, 120 times 0, 1, 314 times 0, 1, 511 times 0, 1, 750 times 0) [i] based on linear OA(919, 734, F9, 7) (dual of [734, 715, 8]-code), using
- construction XX applied to C1 = C([727,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([727,5]) [i] based on
- linear OA(916, 728, F9, 6) (dual of [728, 712, 7]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,4}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(916, 728, F9, 6) (dual of [728, 712, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(919, 728, F9, 7) (dual of [728, 709, 8]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(913, 728, F9, 5) (dual of [728, 715, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [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([727,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([727,5]) [i] based on
- 1720 step Varšamov–Edel lengthening with (ri) = (1, 20 times 0, 1, 120 times 0, 1, 314 times 0, 1, 511 times 0, 1, 750 times 0) [i] based on linear OA(919, 734, F9, 7) (dual of [734, 715, 8]-code), using
(24−7, 24, 4700595)-Net in Base 9 — Upper bound on s
There is no (17, 24, 4700596)-net in base 9, because
- 1 times m-reduction [i] would yield (17, 23, 4700596)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 8862 941275 574159 821409 > 923 [i]