Best Known (70−33, 70, s)-Nets in Base 64
(70−33, 70, 513)-Net over F64 — Constructive and digital
Digital (37, 70, 513)-net over F64, using
- t-expansion [i] based on digital (28, 70, 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
(70−33, 70, 517)-Net in Base 64 — Constructive
(37, 70, 517)-net in base 64, using
- (u, u+v)-construction [i] based on
- (7, 23, 258)-net in base 64, using
- 1 times m-reduction [i] based on (7, 24, 258)-net in base 64, using
- base change [i] based on digital (1, 18, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- base change [i] based on digital (1, 18, 258)-net over F256, using
- 1 times m-reduction [i] based on (7, 24, 258)-net in base 64, using
- (14, 47, 259)-net in base 64, using
- 1 times m-reduction [i] based on (14, 48, 259)-net in base 64, using
- base change [i] based on digital (2, 36, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- base change [i] based on digital (2, 36, 259)-net over F256, using
- 1 times m-reduction [i] based on (14, 48, 259)-net in base 64, using
- (7, 23, 258)-net in base 64, using
(70−33, 70, 2057)-Net over F64 — Digital
Digital (37, 70, 2057)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6470, 2057, F64, 2, 33) (dual of [(2057, 2), 4044, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6470, 4114, F64, 33) (dual of [4114, 4044, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- linear OA(6465, 4097, F64, 33) (dual of [4097, 4032, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(6453, 4097, F64, 27) (dual of [4097, 4044, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(645, 17, F64, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,64)), using
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- Reed–Solomon code RS(59,64) [i]
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- OOA 2-folding [i] based on linear OA(6470, 4114, F64, 33) (dual of [4114, 4044, 34]-code), using
(70−33, 70, 6642763)-Net in Base 64 — Upper bound on s
There is no (37, 70, 6642764)-net in base 64, because
- 1 times m-reduction [i] would yield (37, 69, 6642764)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 42307 609046 225899 100707 469579 294198 602236 677796 569877 556739 463976 363516 274713 647217 905572 502908 309455 396825 149348 864105 304660 > 6469 [i]