Best Known (97, 97+32, s)-Nets in Base 16
(97, 97+32, 8192)-Net over F16 — Constructive and digital
Digital (97, 129, 8192)-net over F16, using
- 1 times m-reduction [i] based on digital (97, 130, 8192)-net over F16, using
- net defined by OOA [i] based on linear OOA(16130, 8192, F16, 33, 33) (dual of [(8192, 33), 270206, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(16130, 131073, F16, 33) (dual of [131073, 130943, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(16130, 131074, F16, 33) (dual of [131074, 130944, 34]-code), using
- trace code [i] based on linear OA(25665, 65537, F256, 33) (dual of [65537, 65472, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- trace code [i] based on linear OA(25665, 65537, F256, 33) (dual of [65537, 65472, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(16130, 131074, F16, 33) (dual of [131074, 130944, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(16130, 131073, F16, 33) (dual of [131073, 130943, 34]-code), using
- net defined by OOA [i] based on linear OOA(16130, 8192, F16, 33, 33) (dual of [(8192, 33), 270206, 34]-NRT-code), using
(97, 97+32, 110208)-Net over F16 — Digital
Digital (97, 129, 110208)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16129, 110208, F16, 32) (dual of [110208, 110079, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(16129, 131083, F16, 32) (dual of [131083, 130954, 33]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16128, 131082, F16, 32) (dual of [131082, 130954, 33]-code), using
- trace code [i] based on linear OA(25664, 65541, F256, 32) (dual of [65541, 65477, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(29) [i] based on
- linear OA(25663, 65536, F256, 32) (dual of [65536, 65473, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(25659, 65536, F256, 30) (dual of [65536, 65477, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(31) ⊂ Ce(29) [i] based on
- trace code [i] based on linear OA(25664, 65541, F256, 32) (dual of [65541, 65477, 33]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16128, 131082, F16, 32) (dual of [131082, 130954, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(16129, 131083, F16, 32) (dual of [131083, 130954, 33]-code), using
(97, 97+32, large)-Net in Base 16 — Upper bound on s
There is no (97, 129, large)-net in base 16, because
- 30 times m-reduction [i] would yield (97, 99, large)-net in base 16, but