Best Known (58, 58+34, s)-Nets in Base 9
(58, 58+34, 344)-Net over F9 — Constructive and digital
Digital (58, 92, 344)-net over F9, using
- 10 times m-reduction [i] based on digital (58, 102, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 51, 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, 51, 172)-net over F81, using
(58, 58+34, 771)-Net over F9 — Digital
Digital (58, 92, 771)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(992, 771, F9, 34) (dual of [771, 679, 35]-code), using
- 36 step Varšamov–Edel lengthening with (ri) = (1, 35 times 0) [i] based on linear OA(991, 734, F9, 34) (dual of [734, 643, 35]-code), using
- construction XX applied to C1 = C([727,31]), C2 = C([0,32]), C3 = C1 + C2 = C([0,31]), and C∩ = C1 ∩ C2 = C([727,32]) [i] based on
- linear OA(988, 728, F9, 33) (dual of [728, 640, 34]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,31}, and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(988, 728, F9, 33) (dual of [728, 640, 34]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,32], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(991, 728, F9, 34) (dual of [728, 637, 35]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,32}, and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(985, 728, F9, 32) (dual of [728, 643, 33]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,31], and designed minimum distance d ≥ |I|+1 = 33 [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([727,31]), C2 = C([0,32]), C3 = C1 + C2 = C([0,31]), and C∩ = C1 ∩ C2 = C([727,32]) [i] based on
- 36 step Varšamov–Edel lengthening with (ri) = (1, 35 times 0) [i] based on linear OA(991, 734, F9, 34) (dual of [734, 643, 35]-code), using
(58, 58+34, 130905)-Net in Base 9 — Upper bound on s
There is no (58, 92, 130906)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 6170 845399 030958 444617 975141 463010 838092 340585 352105 077354 879686 567668 140780 447730 992593 > 992 [i]