Best Known (8, 17, s)-Nets in Base 32
(8, 17, 256)-Net over F32 — Constructive and digital
Digital (8, 17, 256)-net over F32, using
- net defined by OOA [i] based on linear OOA(3217, 256, F32, 9, 9) (dual of [(256, 9), 2287, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(3217, 1025, F32, 9) (dual of [1025, 1008, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(3217, 1025, F32, 9) (dual of [1025, 1008, 10]-code), using
(8, 17, 258)-Net in Base 32 — Constructive
(8, 17, 258)-net in base 32, using
- 1 times m-reduction [i] based on (8, 18, 258)-net in base 32, using
- base change [i] based on (5, 15, 258)-net in base 64, using
- 1 times m-reduction [i] based on (5, 16, 258)-net in base 64, using
- base change [i] based on digital (1, 12, 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, 12, 258)-net over F256, using
- 1 times m-reduction [i] based on (5, 16, 258)-net in base 64, using
- base change [i] based on (5, 15, 258)-net in base 64, using
(8, 17, 513)-Net over F32 — Digital
Digital (8, 17, 513)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3217, 513, F32, 2, 9) (dual of [(513, 2), 1009, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3217, 1026, F32, 9) (dual of [1026, 1009, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(3217, 1024, F32, 9) (dual of [1024, 1007, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(3215, 1024, F32, 8) (dual of [1024, 1009, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- OOA 2-folding [i] based on linear OA(3217, 1026, F32, 9) (dual of [1026, 1009, 10]-code), using
(8, 17, 74865)-Net in Base 32 — Upper bound on s
There is no (8, 17, 74866)-net in base 32, because
- 1 times m-reduction [i] would yield (8, 16, 74866)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 1 208959 778727 582559 951956 > 3216 [i]