Best Known (31, 45, s)-Nets in Base 9
(31, 45, 364)-Net over F9 — Constructive and digital
Digital (31, 45, 364)-net over F9, using
- 1 times m-reduction [i] based on digital (31, 46, 364)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (7, 14, 164)-net over F9, using
- trace code for nets [i] based on digital (0, 7, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 7, 82)-net over F81, using
- digital (17, 32, 200)-net over F9, using
- trace code for nets [i] based on digital (1, 16, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 16, 100)-net over F81, using
- digital (7, 14, 164)-net over F9, using
- (u, u+v)-construction [i] based on
(31, 45, 1431)-Net over F9 — Digital
Digital (31, 45, 1431)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(945, 1431, F9, 14) (dual of [1431, 1386, 15]-code), using
- 689 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 15 times 0, 1, 40 times 0, 1, 87 times 0, 1, 138 times 0, 1, 180 times 0, 1, 219 times 0) [i] based on linear OA(937, 734, F9, 14) (dual of [734, 697, 15]-code), using
- construction XX applied to C1 = C([727,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([727,12]) [i] based on
- linear OA(934, 728, F9, 13) (dual of [728, 694, 14]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,11}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(934, 728, F9, 13) (dual of [728, 694, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(937, 728, F9, 14) (dual of [728, 691, 15]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,12}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(931, 728, F9, 12) (dual of [728, 697, 13]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [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([727,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([727,12]) [i] based on
- 689 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 15 times 0, 1, 40 times 0, 1, 87 times 0, 1, 138 times 0, 1, 180 times 0, 1, 219 times 0) [i] based on linear OA(937, 734, F9, 14) (dual of [734, 697, 15]-code), using
(31, 45, 575760)-Net in Base 9 — Upper bound on s
There is no (31, 45, 575761)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 8 728016 136483 838393 245083 652663 556554 679545 > 945 [i]