Best Known (91−39, 91, s)-Nets in Base 64
(91−39, 91, 641)-Net over F64 — Constructive and digital
Digital (52, 91, 641)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (5, 24, 128)-net over F64, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 5 and N(F) ≥ 128, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- digital (28, 67, 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 (5, 24, 128)-net over F64, using
(91−39, 91, 862)-Net in Base 64 — Constructive
(52, 91, 862)-net in base 64, using
- base change [i] based on digital (39, 78, 862)-net over F128, using
- 1281 times duplication [i] based on digital (38, 77, 862)-net over F128, using
- net defined by OOA [i] based on linear OOA(12877, 862, F128, 39, 39) (dual of [(862, 39), 33541, 40]-NRT-code), using
- OOA 19-folding and stacking with additional row [i] based on linear OA(12877, 16379, F128, 39) (dual of [16379, 16302, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(12877, 16384, F128, 39) (dual of [16384, 16307, 40]-code), using
- an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- discarding factors / shortening the dual code based on linear OA(12877, 16384, F128, 39) (dual of [16384, 16307, 40]-code), using
- OOA 19-folding and stacking with additional row [i] based on linear OA(12877, 16379, F128, 39) (dual of [16379, 16302, 40]-code), using
- net defined by OOA [i] based on linear OOA(12877, 862, F128, 39, 39) (dual of [(862, 39), 33541, 40]-NRT-code), using
- 1281 times duplication [i] based on digital (38, 77, 862)-net over F128, using
(91−39, 91, 5096)-Net over F64 — Digital
Digital (52, 91, 5096)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6491, 5096, F64, 39) (dual of [5096, 5005, 40]-code), using
- 984 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 19 times 0, 1, 38 times 0, 1, 76 times 0, 1, 147 times 0, 1, 265 times 0, 1, 418 times 0) [i] based on linear OA(6477, 4098, F64, 39) (dual of [4098, 4021, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(37) [i] based on
- linear OA(6477, 4096, F64, 39) (dual of [4096, 4019, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- 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(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(38) ⊂ Ce(37) [i] based on
- 984 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 19 times 0, 1, 38 times 0, 1, 76 times 0, 1, 147 times 0, 1, 265 times 0, 1, 418 times 0) [i] based on linear OA(6477, 4098, F64, 39) (dual of [4098, 4021, 40]-code), using
(91−39, 91, large)-Net in Base 64 — Upper bound on s
There is no (52, 91, large)-net in base 64, because
- 37 times m-reduction [i] would yield (52, 54, large)-net in base 64, but