Best Known (33, 33+34, s)-Nets in Base 64
(33, 33+34, 513)-Net over F64 — Constructive and digital
Digital (33, 67, 513)-net over F64, using
- t-expansion [i] based on 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
(33, 33+34, 1349)-Net over F64 — Digital
Digital (33, 67, 1349)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6467, 1349, F64, 3, 34) (dual of [(1349, 3), 3980, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6467, 1366, F64, 3, 34) (dual of [(1366, 3), 4031, 35]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6467, 4098, F64, 34) (dual of [4098, 4031, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- linear OA(6467, 4096, F64, 34) (dual of [4096, 4029, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(6465, 4096, F64, 33) (dual of [4096, 4031, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [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(33) ⊂ Ce(32) [i] based on
- OOA 3-folding [i] based on linear OA(6467, 4098, F64, 34) (dual of [4098, 4031, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(6467, 1366, F64, 3, 34) (dual of [(1366, 3), 4031, 35]-NRT-code), using
(33, 33+34, 1496498)-Net in Base 64 — Upper bound on s
There is no (33, 67, 1496499)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 10 329063 593833 899303 650360 649283 287942 806630 708421 493967 708892 579079 709529 089249 455221 217982 964073 773059 281894 390567 708930 > 6467 [i]