Best Known (32, 32+23, s)-Nets in Base 64
(32, 32+23, 513)-Net over F64 — Constructive and digital
Digital (32, 55, 513)-net over F64, using
- t-expansion [i] based on digital (28, 55, 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, 32+23, 1490)-Net in Base 64 — Constructive
(32, 55, 1490)-net in base 64, using
- net defined by OOA [i] based on OOA(6455, 1490, S64, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(6455, 16391, S64, 23), using
- discarding factors based on OA(6455, 16392, S64, 23), using
- discarding parts of the base [i] based on linear OA(12847, 16392, F128, 23) (dual of [16392, 16345, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(12845, 16384, F128, 23) (dual of [16384, 16339, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(12839, 16384, F128, 20) (dual of [16384, 16345, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(1282, 8, F128, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- discarding parts of the base [i] based on linear OA(12847, 16392, F128, 23) (dual of [16392, 16345, 24]-code), using
- discarding factors based on OA(6455, 16392, S64, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(6455, 16391, S64, 23), using
(32, 32+23, 4919)-Net over F64 — Digital
Digital (32, 55, 4919)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6455, 4919, F64, 23) (dual of [4919, 4864, 24]-code), using
- 811 step Varšamov–Edel lengthening with (ri) = (5, 0, 0, 1, 10 times 0, 1, 30 times 0, 1, 84 times 0, 1, 212 times 0, 1, 467 times 0) [i] based on linear OA(6445, 4098, F64, 23) (dual of [4098, 4053, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- 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(6443, 4096, F64, 22) (dual of [4096, 4053, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(22) ⊂ Ce(21) [i] based on
- 811 step Varšamov–Edel lengthening with (ri) = (5, 0, 0, 1, 10 times 0, 1, 30 times 0, 1, 84 times 0, 1, 212 times 0, 1, 467 times 0) [i] based on linear OA(6445, 4098, F64, 23) (dual of [4098, 4053, 24]-code), using
(32, 32+23, large)-Net in Base 64 — Upper bound on s
There is no (32, 55, large)-net in base 64, because
- 21 times m-reduction [i] would yield (32, 34, large)-net in base 64, but