Best Known (81−32, 81, s)-Nets in Base 64
(81−32, 81, 641)-Net over F64 — Constructive and digital
Digital (49, 81, 641)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (5, 21, 128)-net over F64, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 5 and N(F) ≥ 128, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- digital (28, 60, 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
- digital (5, 21, 128)-net over F64, using
(81−32, 81, 1025)-Net in Base 64 — Constructive
(49, 81, 1025)-net in base 64, using
- 1 times m-reduction [i] based on (49, 82, 1025)-net in base 64, using
- net defined by OOA [i] based on OOA(6482, 1025, S64, 33, 33), using
- OOA 16-folding and stacking with additional row [i] based on OA(6482, 16401, S64, 33), using
- discarding factors based on OA(6482, 16402, S64, 33), using
- discarding parts of the base [i] based on linear OA(12870, 16402, F128, 33) (dual of [16402, 16332, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- linear OA(12865, 16385, F128, 33) (dual of [16385, 16320, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(12853, 16385, F128, 27) (dual of [16385, 16332, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(1285, 17, F128, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- discarding parts of the base [i] based on linear OA(12870, 16402, F128, 33) (dual of [16402, 16332, 34]-code), using
- discarding factors based on OA(6482, 16402, S64, 33), using
- OOA 16-folding and stacking with additional row [i] based on OA(6482, 16401, S64, 33), using
- net defined by OOA [i] based on OOA(6482, 1025, S64, 33, 33), using
(81−32, 81, 10345)-Net over F64 — Digital
Digital (49, 81, 10345)-net over F64, using
(81−32, 81, large)-Net in Base 64 — Upper bound on s
There is no (49, 81, large)-net in base 64, because
- 30 times m-reduction [i] would yield (49, 51, large)-net in base 64, but