Best Known (56, 88, s)-Nets in Base 9
(56, 88, 344)-Net over F9 — Constructive and digital
Digital (56, 88, 344)-net over F9, using
- 10 times m-reduction [i] based on digital (56, 98, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 49, 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, 49, 172)-net over F81, using
(56, 88, 810)-Net over F9 — Digital
Digital (56, 88, 810)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(988, 810, F9, 32) (dual of [810, 722, 33]-code), using
- 73 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0, 1, 22 times 0, 1, 45 times 0) [i] based on linear OA(985, 734, F9, 32) (dual of [734, 649, 33]-code), using
- construction XX applied to C1 = C([727,29]), C2 = C([0,30]), C3 = C1 + C2 = C([0,29]), and C∩ = C1 ∩ C2 = C([727,30]) [i] based on
- linear OA(982, 728, F9, 31) (dual of [728, 646, 32]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,29}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(982, 728, F9, 31) (dual of [728, 646, 32]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,30], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(985, 728, F9, 32) (dual of [728, 643, 33]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,30}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(979, 728, F9, 30) (dual of [728, 649, 31]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,29], and designed minimum distance d ≥ |I|+1 = 31 [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([727,29]), C2 = C([0,30]), C3 = C1 + C2 = C([0,29]), and C∩ = C1 ∩ C2 = C([727,30]) [i] based on
- 73 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0, 1, 22 times 0, 1, 45 times 0) [i] based on linear OA(985, 734, F9, 32) (dual of [734, 649, 33]-code), using
(56, 88, 150575)-Net in Base 9 — Upper bound on s
There is no (56, 88, 150576)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 940482 404244 356337 669177 571339 132263 021343 570600 761982 802983 137783 000485 246242 580481 > 988 [i]