Best Known (54, 54+32, s)-Nets in Base 9
(54, 54+32, 344)-Net over F9 — Constructive and digital
Digital (54, 86, 344)-net over F9, using
- 8 times m-reduction [i] based on digital (54, 94, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 47, 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, 47, 172)-net over F81, using
(54, 54+32, 739)-Net over F9 — Digital
Digital (54, 86, 739)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(986, 739, F9, 32) (dual of [739, 653, 33]-code), using
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 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
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0) [i] based on linear OA(985, 734, F9, 32) (dual of [734, 649, 33]-code), using
(54, 54+32, 114410)-Net in Base 9 — Upper bound on s
There is no (54, 86, 114411)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 11611 321314 268101 048902 320319 834221 888174 647448 338282 555314 223590 617685 856578 048385 > 986 [i]