Best Known (21, 32, s)-Nets in Base 9
(21, 32, 328)-Net over F9 — Constructive and digital
Digital (21, 32, 328)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (5, 10, 164)-net over F9, using
- trace code for nets [i] based on digital (0, 5, 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, 5, 82)-net over F81, using
- digital (11, 22, 164)-net over F9, using
- trace code for nets [i] based on digital (0, 11, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81 (see above)
- trace code for nets [i] based on digital (0, 11, 82)-net over F81, using
- digital (5, 10, 164)-net over F9, using
(21, 32, 784)-Net over F9 — Digital
Digital (21, 32, 784)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(932, 784, F9, 11) (dual of [784, 752, 12]-code), using
- 48 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 11 times 0, 1, 32 times 0) [i] based on linear OA(928, 732, F9, 11) (dual of [732, 704, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(928, 729, F9, 11) (dual of [729, 701, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(925, 729, F9, 10) (dual of [729, 704, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- 48 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 11 times 0, 1, 32 times 0) [i] based on linear OA(928, 732, F9, 11) (dual of [732, 704, 12]-code), using
(21, 32, 268562)-Net in Base 9 — Upper bound on s
There is no (21, 32, 268563)-net in base 9, because
- 1 times m-reduction [i] would yield (21, 31, 268563)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 381523 076206 276997 118027 423225 > 931 [i]