Best Known (126−33, 126, s)-Nets in Base 16
(126−33, 126, 4097)-Net over F16 — Constructive and digital
Digital (93, 126, 4097)-net over F16, using
- 161 times duplication [i] based on digital (92, 125, 4097)-net over F16, using
- net defined by OOA [i] based on linear OOA(16125, 4097, F16, 33, 33) (dual of [(4097, 33), 135076, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(16125, 65553, F16, 33) (dual of [65553, 65428, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(27) [i] based on
- linear OA(16121, 65536, F16, 33) (dual of [65536, 65415, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(16105, 65536, F16, 28) (dual of [65536, 65431, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(164, 17, F16, 4) (dual of [17, 13, 5]-code or 17-arc in PG(3,16)), using
- extended Reed–Solomon code RSe(13,16) [i]
- construction X applied to Ce(32) ⊂ Ce(27) [i] based on
- OOA 16-folding and stacking with additional row [i] based on linear OA(16125, 65553, F16, 33) (dual of [65553, 65428, 34]-code), using
- net defined by OOA [i] based on linear OOA(16125, 4097, F16, 33, 33) (dual of [(4097, 33), 135076, 34]-NRT-code), using
(126−33, 126, 59312)-Net over F16 — Digital
Digital (93, 126, 59312)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16126, 59312, F16, 33) (dual of [59312, 59186, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(16126, 65554, F16, 33) (dual of [65554, 65428, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- 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(16105, 65537, F16, 27) (dual of [65537, 65432, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 168−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(165, 17, F16, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,16)), using
- extended Reed–Solomon code RSe(12,16) [i]
- the expurgated narrow-sense BCH-code C(I) with length 17 | 162−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(16126, 65554, F16, 33) (dual of [65554, 65428, 34]-code), using
(126−33, 126, large)-Net in Base 16 — Upper bound on s
There is no (93, 126, large)-net in base 16, because
- 31 times m-reduction [i] would yield (93, 95, large)-net in base 16, but