Best Known (45, 45+23, s)-Nets in Base 9
(45, 45+23, 344)-Net over F9 — Constructive and digital
Digital (45, 68, 344)-net over F9, using
- 8 times m-reduction [i] based on digital (45, 76, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 38, 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, 38, 172)-net over F81, using
(45, 45+23, 1019)-Net over F9 — Digital
Digital (45, 68, 1019)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(968, 1019, F9, 23) (dual of [1019, 951, 24]-code), using
- 278 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 11 times 0, 1, 30 times 0, 1, 57 times 0, 1, 78 times 0, 1, 93 times 0) [i] based on linear OA(961, 734, F9, 23) (dual of [734, 673, 24]-code), using
- construction XX applied to C1 = C([727,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([727,21]) [i] based on
- 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(958, 728, F9, 22) (dual of [728, 670, 23]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(961, 728, F9, 23) (dual of [728, 667, 24]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,21}, and designed minimum distance d ≥ |I|+1 = 24 [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(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([727,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([727,21]) [i] based on
- 278 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 11 times 0, 1, 30 times 0, 1, 57 times 0, 1, 78 times 0, 1, 93 times 0) [i] based on linear OA(961, 734, F9, 23) (dual of [734, 673, 24]-code), using
(45, 45+23, 398218)-Net in Base 9 — Upper bound on s
There is no (45, 68, 398219)-net in base 9, because
- 1 times m-reduction [i] would yield (45, 67, 398219)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 8595 128371 695964 014553 071266 607351 330900 627485 463788 363308 105545 > 967 [i]