Best Known (37, 37+37, s)-Nets in Base 64
(37, 37+37, 513)-Net over F64 — Constructive and digital
Digital (37, 74, 513)-net over F64, using
- t-expansion [i] based on digital (28, 74, 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
(37, 37+37, 514)-Net in Base 64 — Constructive
(37, 74, 514)-net in base 64, using
- (u, u+v)-construction [i] based on
- (6, 24, 257)-net in base 64, using
- base change [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
- base change [i] based on digital (0, 18, 257)-net over F256, using
- (13, 50, 257)-net in base 64, using
- 2 times m-reduction [i] based on (13, 52, 257)-net in base 64, using
- base change [i] based on digital (0, 39, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- base change [i] based on digital (0, 39, 257)-net over F256, using
- 2 times m-reduction [i] based on (13, 52, 257)-net in base 64, using
- (6, 24, 257)-net in base 64, using
(37, 37+37, 1419)-Net over F64 — Digital
Digital (37, 74, 1419)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6474, 1419, F64, 2, 37) (dual of [(1419, 2), 2764, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6474, 2051, F64, 2, 37) (dual of [(2051, 2), 4028, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6474, 4102, F64, 37) (dual of [4102, 4028, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,17]) [i] based on
- linear OA(6473, 4097, F64, 37) (dual of [4097, 4024, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(6469, 4097, F64, 35) (dual of [4097, 4028, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [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 C([0,18]) ⊂ C([0,17]) [i] based on
- OOA 2-folding [i] based on linear OA(6474, 4102, F64, 37) (dual of [4102, 4028, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(6474, 2051, F64, 2, 37) (dual of [(2051, 2), 4028, 38]-NRT-code), using
(37, 37+37, 2534260)-Net in Base 64 — Upper bound on s
There is no (37, 74, 2534261)-net in base 64, because
- 1 times m-reduction [i] would yield (37, 73, 2534261)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 709806 700711 907153 934248 118360 054700 265215 811899 715399 302392 269044 566724 147000 461192 536700 544646 628909 074736 794100 227761 355875 614490 > 6473 [i]