Best Known (87−30, 87, s)-Nets in Base 64
(87−30, 87, 1026)-Net over F64 — Constructive and digital
Digital (57, 87, 1026)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (14, 29, 585)-net over F64, using
- net defined by OOA [i] based on linear OOA(6429, 585, F64, 15, 15) (dual of [(585, 15), 8746, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6429, 4096, F64, 15) (dual of [4096, 4067, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(6429, 4096, F64, 15) (dual of [4096, 4067, 16]-code), using
- net defined by OOA [i] based on linear OOA(6429, 585, F64, 15, 15) (dual of [(585, 15), 8746, 16]-NRT-code), using
- digital (28, 58, 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 (14, 29, 585)-net over F64, using
(87−30, 87, 4370)-Net in Base 64 — Constructive
(57, 87, 4370)-net in base 64, using
- 1 times m-reduction [i] based on (57, 88, 4370)-net in base 64, using
- base change [i] based on digital (35, 66, 4370)-net over F256, using
- net defined by OOA [i] based on linear OOA(25666, 4370, F256, 31, 31) (dual of [(4370, 31), 135404, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(25666, 65551, F256, 31) (dual of [65551, 65485, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(25666, 65554, F256, 31) (dual of [65554, 65488, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- linear OA(25661, 65537, F256, 31) (dual of [65537, 65476, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2565, 17, F256, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,256)), using
- discarding factors / shortening the dual code based on linear OA(2565, 256, F256, 5) (dual of [256, 251, 6]-code or 256-arc in PG(4,256)), using
- Reed–Solomon code RS(251,256) [i]
- discarding factors / shortening the dual code based on linear OA(2565, 256, F256, 5) (dual of [256, 251, 6]-code or 256-arc in PG(4,256)), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25666, 65554, F256, 31) (dual of [65554, 65488, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(25666, 65551, F256, 31) (dual of [65551, 65485, 32]-code), using
- net defined by OOA [i] based on linear OOA(25666, 4370, F256, 31, 31) (dual of [(4370, 31), 135404, 32]-NRT-code), using
- base change [i] based on digital (35, 66, 4370)-net over F256, using
(87−30, 87, 48579)-Net over F64 — Digital
Digital (57, 87, 48579)-net over F64, using
(87−30, 87, large)-Net in Base 64 — Upper bound on s
There is no (57, 87, large)-net in base 64, because
- 28 times m-reduction [i] would yield (57, 59, large)-net in base 64, but