Best Known (80−35, 80, s)-Nets in Base 64
(80−35, 80, 578)-Net over F64 — Constructive and digital
Digital (45, 80, 578)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 17, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- digital (28, 63, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- digital (0, 17, 65)-net over F64, using
(80−35, 80, 4350)-Net over F64 — Digital
Digital (45, 80, 4350)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6480, 4350, F64, 35) (dual of [4350, 4270, 36]-code), using
- 241 step Varšamov–Edel lengthening with (ri) = (6, 0, 0, 1, 5 times 0, 1, 13 times 0, 1, 31 times 0, 1, 62 times 0, 1, 122 times 0) [i] based on linear OA(6469, 4098, F64, 35) (dual of [4098, 4029, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- linear OA(6469, 4096, F64, 35) (dual of [4096, 4027, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(6467, 4096, F64, 34) (dual of [4096, 4029, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- 241 step Varšamov–Edel lengthening with (ri) = (6, 0, 0, 1, 5 times 0, 1, 13 times 0, 1, 31 times 0, 1, 62 times 0, 1, 122 times 0) [i] based on linear OA(6469, 4098, F64, 35) (dual of [4098, 4029, 36]-code), using
(80−35, 80, large)-Net in Base 64 — Upper bound on s
There is no (45, 80, large)-net in base 64, because
- 33 times m-reduction [i] would yield (45, 47, large)-net in base 64, but