Best Known (72−36, 72, s)-Nets in Base 64
(72−36, 72, 513)-Net over F64 — Constructive and digital
Digital (36, 72, 513)-net over F64, using
- t-expansion [i] based on digital (28, 72, 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
(72−36, 72, 514)-Net in Base 64 — Constructive
(36, 72, 514)-net in base 64, using
- base change [i] based on digital (18, 54, 514)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (0, 18, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- digital (0, 36, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- digital (0, 18, 257)-net over F256, using
- (u, u+v)-construction [i] based on
(72−36, 72, 1401)-Net over F64 — Digital
Digital (36, 72, 1401)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6472, 1401, F64, 2, 36) (dual of [(1401, 2), 2730, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6472, 2050, F64, 2, 36) (dual of [(2050, 2), 4028, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6472, 4100, F64, 36) (dual of [4100, 4028, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(6472, 4101, F64, 36) (dual of [4101, 4029, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(33) [i] based on
- linear OA(6471, 4096, F64, 36) (dual of [4096, 4025, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- 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(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(35) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(6472, 4101, F64, 36) (dual of [4101, 4029, 37]-code), using
- OOA 2-folding [i] based on linear OA(6472, 4100, F64, 36) (dual of [4100, 4028, 37]-code), using
- discarding factors / shortening the dual code based on linear OOA(6472, 2050, F64, 2, 36) (dual of [(2050, 2), 4028, 37]-NRT-code), using
(72−36, 72, 2011441)-Net in Base 64 — Upper bound on s
There is no (36, 72, 2011442)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 11090 682437 102657 880761 392863 968841 065637 968111 338980 627691 336544 976051 191363 443416 543165 961359 594940 947688 854815 072021 872200 797296 > 6472 [i]