Best Known (44, 44+34, s)-Nets in Base 64
(44, 44+34, 513)-Net over F64 — Constructive and digital
Digital (44, 78, 513)-net over F64, using
- t-expansion [i] based on digital (28, 78, 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
(44, 44+34, 548)-Net in Base 64 — Constructive
(44, 78, 548)-net in base 64, using
- (u, u+v)-construction [i] based on
- (10, 27, 260)-net in base 64, using
- 1 times m-reduction [i] based on (10, 28, 260)-net in base 64, using
- base change [i] based on digital (3, 21, 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, 21, 260)-net over F256, using
- 1 times m-reduction [i] based on (10, 28, 260)-net in base 64, using
- (17, 51, 288)-net in base 64, using
- 5 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- base change [i] based on digital (9, 48, 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, 48, 288)-net over F128, using
- 5 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- (10, 27, 260)-net in base 64, using
(44, 44+34, 4381)-Net over F64 — Digital
Digital (44, 78, 4381)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6478, 4381, F64, 34) (dual of [4381, 4303, 35]-code), using
- 272 step Varšamov–Edel lengthening with (ri) = (6, 0, 0, 1, 5 times 0, 1, 15 times 0, 1, 33 times 0, 1, 71 times 0, 1, 140 times 0) [i] based on linear OA(6467, 4098, F64, 34) (dual of [4098, 4031, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- 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(6465, 4096, F64, 33) (dual of [4096, 4031, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [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(33) ⊂ Ce(32) [i] based on
- 272 step Varšamov–Edel lengthening with (ri) = (6, 0, 0, 1, 5 times 0, 1, 15 times 0, 1, 33 times 0, 1, 71 times 0, 1, 140 times 0) [i] based on linear OA(6467, 4098, F64, 34) (dual of [4098, 4031, 35]-code), using
(44, 44+34, large)-Net in Base 64 — Upper bound on s
There is no (44, 78, large)-net in base 64, because
- 32 times m-reduction [i] would yield (44, 46, large)-net in base 64, but