Best Known (49, 49+38, s)-Nets in Base 64
(49, 49+38, 593)-Net over F64 — Constructive and digital
Digital (49, 87, 593)-net over F64, using
- 2 times m-reduction [i] based on digital (49, 89, 593)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 21, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- digital (28, 68, 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 (1, 21, 80)-net over F64, using
- (u, u+v)-construction [i] based on
(49, 49+38, 4443)-Net over F64 — Digital
Digital (49, 87, 4443)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6487, 4443, F64, 38) (dual of [4443, 4356, 39]-code), using
- 333 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 4 times 0, 1, 9 times 0, 1, 21 times 0, 1, 43 times 0, 1, 85 times 0, 1, 163 times 0) [i] based on linear OA(6475, 4098, F64, 38) (dual of [4098, 4023, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- linear OA(6475, 4096, F64, 38) (dual of [4096, 4021, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(6473, 4096, F64, 37) (dual of [4096, 4023, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(37) ⊂ Ce(36) [i] based on
- 333 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 4 times 0, 1, 9 times 0, 1, 21 times 0, 1, 43 times 0, 1, 85 times 0, 1, 163 times 0) [i] based on linear OA(6475, 4098, F64, 38) (dual of [4098, 4023, 39]-code), using
(49, 49+38, large)-Net in Base 64 — Upper bound on s
There is no (49, 87, large)-net in base 64, because
- 36 times m-reduction [i] would yield (49, 51, large)-net in base 64, but