Best Known (40, 71, s)-Nets in Base 64
(40, 71, 513)-Net over F64 — Constructive and digital
Digital (40, 71, 513)-net over F64, using
- t-expansion [i] based on digital (28, 71, 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
(40, 71, 548)-Net in Base 64 — Constructive
(40, 71, 548)-net in base 64, using
- (u, u+v)-construction [i] based on
- (9, 24, 260)-net in base 64, using
- base change [i] based on digital (3, 18, 260)-net over F256, using
- net from sequence [i] based on digital (3, 259)-sequence over F256, using
- base change [i] based on digital (3, 18, 260)-net over F256, using
- (16, 47, 288)-net in base 64, using
- 2 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
- base change [i] based on digital (9, 42, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 42, 288)-net over F128, using
- 2 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
- (9, 24, 260)-net in base 64, using
(40, 71, 4228)-Net over F64 — Digital
Digital (40, 71, 4228)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6471, 4228, F64, 31) (dual of [4228, 4157, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(6471, 4272, F64, 31) (dual of [4272, 4201, 32]-code), using
- 164 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 1, 6 times 0, 1, 17 times 0, 1, 41 times 0, 1, 93 times 0) [i] based on linear OA(6461, 4098, F64, 31) (dual of [4098, 4037, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(29) [i] based on
- linear OA(6461, 4096, F64, 31) (dual of [4096, 4035, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(6459, 4096, F64, 30) (dual of [4096, 4037, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(30) ⊂ Ce(29) [i] based on
- 164 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 1, 6 times 0, 1, 17 times 0, 1, 41 times 0, 1, 93 times 0) [i] based on linear OA(6461, 4098, F64, 31) (dual of [4098, 4037, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(6471, 4272, F64, 31) (dual of [4272, 4201, 32]-code), using
(40, 71, large)-Net in Base 64 — Upper bound on s
There is no (40, 71, large)-net in base 64, because
- 29 times m-reduction [i] would yield (40, 42, large)-net in base 64, but