Best Known (39, 39+22, s)-Nets in Base 9
(39, 39+22, 344)-Net over F9 — Constructive and digital
Digital (39, 61, 344)-net over F9, using
- 3 times m-reduction [i] based on digital (39, 64, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 32, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 32, 172)-net over F81, using
(39, 39+22, 746)-Net over F9 — Digital
Digital (39, 61, 746)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(961, 746, F9, 22) (dual of [746, 685, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(961, 748, F9, 22) (dual of [748, 687, 23]-code), using
- 11 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 7 times 0) [i] based on linear OA(958, 734, F9, 22) (dual of [734, 676, 23]-code), using
- construction XX applied to C1 = C([727,19]), C2 = C([0,20]), C3 = C1 + C2 = C([0,19]), and C∩ = C1 ∩ C2 = C([727,20]) [i] based on
- linear OA(955, 728, F9, 21) (dual of [728, 673, 22]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,19}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(955, 728, F9, 21) (dual of [728, 673, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(958, 728, F9, 22) (dual of [728, 670, 23]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,20}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(952, 728, F9, 20) (dual of [728, 676, 21]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [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([727,19]), C2 = C([0,20]), C3 = C1 + C2 = C([0,19]), and C∩ = C1 ∩ C2 = C([727,20]) [i] based on
- 11 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 7 times 0) [i] based on linear OA(958, 734, F9, 22) (dual of [734, 676, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(961, 748, F9, 22) (dual of [748, 687, 23]-code), using
(39, 39+22, 120118)-Net in Base 9 — Upper bound on s
There is no (39, 61, 120119)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 16173 963377 864732 345194 916231 070964 779792 784446 221279 779305 > 961 [i]