Best Known (91, 91+30, s)-Nets in Base 16
(91, 91+30, 8738)-Net over F16 — Constructive and digital
Digital (91, 121, 8738)-net over F16, using
- 1 times m-reduction [i] based on digital (91, 122, 8738)-net over F16, using
- net defined by OOA [i] based on linear OOA(16122, 8738, F16, 31, 31) (dual of [(8738, 31), 270756, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(16122, 131071, F16, 31) (dual of [131071, 130949, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(16122, 131074, F16, 31) (dual of [131074, 130952, 32]-code), using
- trace code [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]
- trace code [i] based on linear OA(25661, 65537, F256, 31) (dual of [65537, 65476, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(16122, 131074, F16, 31) (dual of [131074, 130952, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(16122, 131071, F16, 31) (dual of [131071, 130949, 32]-code), using
- net defined by OOA [i] based on linear OOA(16122, 8738, F16, 31, 31) (dual of [(8738, 31), 270756, 32]-NRT-code), using
(91, 91+30, 108986)-Net over F16 — Digital
Digital (91, 121, 108986)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16121, 108986, F16, 30) (dual of [108986, 108865, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(16121, 131083, F16, 30) (dual of [131083, 130962, 31]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16120, 131082, F16, 30) (dual of [131082, 130962, 31]-code), using
- trace code [i] based on linear OA(25660, 65541, F256, 30) (dual of [65541, 65481, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(27) [i] based on
- 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(25655, 65536, F256, 28) (dual of [65536, 65481, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(29) ⊂ Ce(27) [i] based on
- trace code [i] based on linear OA(25660, 65541, F256, 30) (dual of [65541, 65481, 31]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16120, 131082, F16, 30) (dual of [131082, 130962, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(16121, 131083, F16, 30) (dual of [131083, 130962, 31]-code), using
(91, 91+30, large)-Net in Base 16 — Upper bound on s
There is no (91, 121, large)-net in base 16, because
- 28 times m-reduction [i] would yield (91, 93, large)-net in base 16, but