Best Known (130−35, 130, s)-Nets in Base 16
(130−35, 130, 3855)-Net over F16 — Constructive and digital
Digital (95, 130, 3855)-net over F16, using
- 161 times duplication [i] based on digital (94, 129, 3855)-net over F16, using
- net defined by OOA [i] based on linear OOA(16129, 3855, F16, 35, 35) (dual of [(3855, 35), 134796, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(16129, 65536, F16, 35) (dual of [65536, 65407, 36]-code), using
- an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- OOA 17-folding and stacking with additional row [i] based on linear OA(16129, 65536, F16, 35) (dual of [65536, 65407, 36]-code), using
- net defined by OOA [i] based on linear OOA(16129, 3855, F16, 35, 35) (dual of [(3855, 35), 134796, 36]-NRT-code), using
(130−35, 130, 44679)-Net over F16 — Digital
Digital (95, 130, 44679)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16130, 44679, F16, 35) (dual of [44679, 44549, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(16130, 65546, F16, 35) (dual of [65546, 65416, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,16]) [i] based on
- linear OA(16129, 65537, F16, 35) (dual of [65537, 65408, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 168−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(16121, 65537, F16, 33) (dual of [65537, 65416, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 168−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(161, 9, F16, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,17]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(16130, 65546, F16, 35) (dual of [65546, 65416, 36]-code), using
(130−35, 130, large)-Net in Base 16 — Upper bound on s
There is no (95, 130, large)-net in base 16, because
- 33 times m-reduction [i] would yield (95, 97, large)-net in base 16, but