Best Known (36, 36+13, s)-Nets in Base 16
(36, 36+13, 10923)-Net over F16 — Constructive and digital
Digital (36, 49, 10923)-net over F16, using
- net defined by OOA [i] based on linear OOA(1649, 10923, F16, 13, 13) (dual of [(10923, 13), 141950, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1649, 65539, F16, 13) (dual of [65539, 65490, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(1649, 65540, F16, 13) (dual of [65540, 65491, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(1649, 65536, F16, 13) (dual of [65536, 65487, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(1645, 65536, F16, 12) (dual of [65536, 65491, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(160, 4, F16, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(1649, 65540, F16, 13) (dual of [65540, 65491, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1649, 65539, F16, 13) (dual of [65539, 65490, 14]-code), using
(36, 36+13, 58780)-Net over F16 — Digital
Digital (36, 49, 58780)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1649, 58780, F16, 13) (dual of [58780, 58731, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(1649, 65536, F16, 13) (dual of [65536, 65487, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(1649, 65536, F16, 13) (dual of [65536, 65487, 14]-code), using
(36, 36+13, large)-Net in Base 16 — Upper bound on s
There is no (36, 49, large)-net in base 16, because
- 11 times m-reduction [i] would yield (36, 38, large)-net in base 16, but