Best Known (35, 35+25, s)-Nets in Base 64
(35, 35+25, 513)-Net over F64 — Constructive and digital
Digital (35, 60, 513)-net over F64, using
- t-expansion [i] based on digital (28, 60, 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
(35, 35+25, 1365)-Net in Base 64 — Constructive
(35, 60, 1365)-net in base 64, using
- 642 times duplication [i] based on (33, 58, 1365)-net in base 64, using
- net defined by OOA [i] based on OOA(6458, 1365, S64, 25, 25), using
- OOA 12-folding and stacking with additional row [i] based on OA(6458, 16381, S64, 25), using
- discarding factors based on OA(6458, 16386, S64, 25), using
- discarding parts of the base [i] based on linear OA(12849, 16386, F128, 25) (dual of [16386, 16337, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] 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]
- 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(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- discarding parts of the base [i] based on linear OA(12849, 16386, F128, 25) (dual of [16386, 16337, 26]-code), using
- discarding factors based on OA(6458, 16386, S64, 25), using
- OOA 12-folding and stacking with additional row [i] based on OA(6458, 16381, S64, 25), using
- net defined by OOA [i] based on OOA(6458, 1365, S64, 25, 25), using
(35, 35+25, 5215)-Net over F64 — Digital
Digital (35, 60, 5215)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6460, 5215, F64, 25) (dual of [5215, 5155, 26]-code), using
- 1106 step Varšamov–Edel lengthening with (ri) = (5, 0, 0, 1, 9 times 0, 1, 24 times 0, 1, 61 times 0, 1, 140 times 0, 1, 298 times 0, 1, 565 times 0) [i] based on linear OA(6449, 4098, F64, 25) (dual of [4098, 4049, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- linear OA(6449, 4096, F64, 25) (dual of [4096, 4047, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- 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(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(24) ⊂ Ce(23) [i] based on
- 1106 step Varšamov–Edel lengthening with (ri) = (5, 0, 0, 1, 9 times 0, 1, 24 times 0, 1, 61 times 0, 1, 140 times 0, 1, 298 times 0, 1, 565 times 0) [i] based on linear OA(6449, 4098, F64, 25) (dual of [4098, 4049, 26]-code), using
(35, 35+25, large)-Net in Base 64 — Upper bound on s
There is no (35, 60, large)-net in base 64, because
- 23 times m-reduction [i] would yield (35, 37, large)-net in base 64, but