Best Known (32, 56, s)-Nets in Base 64
(32, 56, 513)-Net over F64 — Constructive and digital
Digital (32, 56, 513)-net over F64, using
- t-expansion [i] based on digital (28, 56, 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
(32, 56, 1365)-Net in Base 64 — Constructive
(32, 56, 1365)-net in base 64, using
- base change [i] based on digital (24, 48, 1365)-net over F128, using
- 1 times m-reduction [i] based on digital (24, 49, 1365)-net over F128, using
- net defined by OOA [i] based on linear OOA(12849, 1365, F128, 25, 25) (dual of [(1365, 25), 34076, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12849, 16381, F128, 25) (dual of [16381, 16332, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(12849, 16384, F128, 25) (dual of [16384, 16335, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(12849, 16384, F128, 25) (dual of [16384, 16335, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12849, 16381, F128, 25) (dual of [16381, 16332, 26]-code), using
- net defined by OOA [i] based on linear OOA(12849, 1365, F128, 25, 25) (dual of [(1365, 25), 34076, 26]-NRT-code), using
- 1 times m-reduction [i] based on digital (24, 49, 1365)-net over F128, using
(32, 56, 4400)-Net over F64 — Digital
Digital (32, 56, 4400)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6456, 4400, F64, 24) (dual of [4400, 4344, 25]-code), using
- 293 step Varšamov–Edel lengthening with (ri) = (5, 0, 0, 1, 10 times 0, 1, 28 times 0, 1, 74 times 0, 1, 174 times 0) [i] based on linear OA(6447, 4098, F64, 24) (dual of [4098, 4051, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- linear OA(6447, 4096, F64, 24) (dual of [4096, 4049, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(6445, 4096, F64, 23) (dual of [4096, 4051, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(23) ⊂ Ce(22) [i] based on
- 293 step Varšamov–Edel lengthening with (ri) = (5, 0, 0, 1, 10 times 0, 1, 28 times 0, 1, 74 times 0, 1, 174 times 0) [i] based on linear OA(6447, 4098, F64, 24) (dual of [4098, 4051, 25]-code), using
(32, 56, large)-Net in Base 64 — Upper bound on s
There is no (32, 56, large)-net in base 64, because
- 22 times m-reduction [i] would yield (32, 34, large)-net in base 64, but