Best Known (38, 38+37, s)-Nets in Base 64
(38, 38+37, 513)-Net over F64 — Constructive and digital
Digital (38, 75, 513)-net over F64, using
- t-expansion [i] based on digital (28, 75, 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
(38, 38+37, 515)-Net in Base 64 — Constructive
(38, 75, 515)-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
- (14, 51, 258)-net in base 64, using
- 1 times m-reduction [i] based on (14, 52, 258)-net in base 64, using
- base change [i] based on digital (1, 39, 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, 39, 258)-net over F256, using
- 1 times m-reduction [i] based on (14, 52, 258)-net in base 64, using
- (6, 24, 257)-net in base 64, using
(38, 38+37, 1606)-Net over F64 — Digital
Digital (38, 75, 1606)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6475, 1606, F64, 2, 37) (dual of [(1606, 2), 3137, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6475, 2052, F64, 2, 37) (dual of [(2052, 2), 4029, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6475, 4104, F64, 37) (dual of [4104, 4029, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(33) [i] based on
- linear OA(6473, 4096, F64, 37) (dual of [4096, 4023, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(36) ⊂ Ce(33) [i] based on
- OOA 2-folding [i] based on linear OA(6475, 4104, F64, 37) (dual of [4104, 4029, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(6475, 2052, F64, 2, 37) (dual of [(2052, 2), 4029, 38]-NRT-code), using
(38, 38+37, 3192970)-Net in Base 64 — Upper bound on s
There is no (38, 75, 3192971)-net in base 64, because
- 1 times m-reduction [i] would yield (38, 74, 3192971)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 45 427612 605019 727827 058966 148882 344059 756876 690212 200166 160867 595361 529888 773938 748243 304557 980756 559626 586619 949363 540388 812711 132104 > 6474 [i]