Best Known (58−24, 58, s)-Nets in Base 64
(58−24, 58, 513)-Net over F64 — Constructive and digital
Digital (34, 58, 513)-net over F64, using
- t-expansion [i] based on digital (28, 58, 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
(58−24, 58, 1366)-Net in Base 64 — Constructive
(34, 58, 1366)-net in base 64, using
- net defined by OOA [i] based on OOA(6458, 1366, S64, 24, 24), using
- OA 12-folding and stacking [i] based on OA(6458, 16392, S64, 24), using
- discarding parts of the base [i] based on linear OA(12849, 16392, F128, 24) (dual of [16392, 16343, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- linear OA(12847, 16384, F128, 24) (dual of [16384, 16337, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(23) ⊂ Ce(20) [i] based on
- discarding parts of the base [i] based on linear OA(12849, 16392, F128, 24) (dual of [16392, 16343, 25]-code), using
- OA 12-folding and stacking [i] based on OA(6458, 16392, S64, 24), using
(58−24, 58, 5453)-Net over F64 — Digital
Digital (34, 58, 5453)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6458, 5453, F64, 24) (dual of [5453, 5395, 25]-code), using
- 1344 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, 1, 371 times 0, 1, 678 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
- 1344 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, 1, 371 times 0, 1, 678 times 0) [i] based on linear OA(6447, 4098, F64, 24) (dual of [4098, 4051, 25]-code), using
(58−24, 58, large)-Net in Base 64 — Upper bound on s
There is no (34, 58, large)-net in base 64, because
- 22 times m-reduction [i] would yield (34, 36, large)-net in base 64, but