Best Known (47, 47+25, s)-Nets in Base 9
(47, 47+25, 344)-Net over F9 — Constructive and digital
Digital (47, 72, 344)-net over F9, using
- 8 times m-reduction [i] based on digital (47, 80, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 40, 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, 40, 172)-net over F81, using
(47, 47+25, 906)-Net over F9 — Digital
Digital (47, 72, 906)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(972, 906, F9, 25) (dual of [906, 834, 26]-code), using
- 167 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 28 times 0, 1, 56 times 0, 1, 73 times 0) [i] based on linear OA(967, 734, F9, 25) (dual of [734, 667, 26]-code), using
- construction XX applied to C1 = C([727,22]), C2 = C([0,23]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([727,23]) [i] based on
- linear OA(964, 728, F9, 24) (dual of [728, 664, 25]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,22}, and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(964, 728, F9, 24) (dual of [728, 664, 25]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,23], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(967, 728, F9, 25) (dual of [728, 661, 26]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,23}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(961, 728, F9, 23) (dual of [728, 667, 24]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [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([727,22]), C2 = C([0,23]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([727,23]) [i] based on
- 167 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 28 times 0, 1, 56 times 0, 1, 73 times 0) [i] based on linear OA(967, 734, F9, 25) (dual of [734, 667, 26]-code), using
(47, 47+25, 292547)-Net in Base 9 — Upper bound on s
There is no (47, 72, 292548)-net in base 9, because
- 1 times m-reduction [i] would yield (47, 71, 292548)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 56 394036 141611 821991 223640 068686 544449 253565 782412 646320 071129 337473 > 971 [i]