Best Known (39−12, 39, s)-Nets in Base 9
(39−12, 39, 364)-Net over F9 — Constructive and digital
Digital (27, 39, 364)-net over F9, using
- 1 times m-reduction [i] based on digital (27, 40, 364)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (6, 12, 164)-net over F9, using
- trace code for nets [i] based on digital (0, 6, 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, 6, 82)-net over F81, using
- digital (15, 28, 200)-net over F9, using
- trace code for nets [i] based on digital (1, 14, 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, 14, 100)-net over F81, using
- digital (6, 12, 164)-net over F9, using
- (u, u+v)-construction [i] based on
(39−12, 39, 1489)-Net over F9 — Digital
Digital (27, 39, 1489)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(939, 1489, F9, 12) (dual of [1489, 1450, 13]-code), using
- 747 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 11 times 0, 1, 32 times 0, 1, 79 times 0, 1, 144 times 0, 1, 207 times 0, 1, 264 times 0) [i] based on linear OA(931, 734, F9, 12) (dual of [734, 703, 13]-code), using
- construction XX applied to C1 = C([727,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([727,10]) [i] based on
- linear OA(928, 728, F9, 11) (dual of [728, 700, 12]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,9}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(928, 728, F9, 11) (dual of [728, 700, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(931, 728, F9, 12) (dual of [728, 697, 13]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,10}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(925, 728, F9, 10) (dual of [728, 703, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [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([727,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([727,10]) [i] based on
- 747 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 11 times 0, 1, 32 times 0, 1, 79 times 0, 1, 144 times 0, 1, 207 times 0, 1, 264 times 0) [i] based on linear OA(931, 734, F9, 12) (dual of [734, 703, 13]-code), using
(39−12, 39, 596631)-Net in Base 9 — Upper bound on s
There is no (27, 39, 596632)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 16 423317 026993 564036 178001 000344 940929 > 939 [i]