Best Known (15, 21, s)-Nets in Base 16
(15, 21, 21846)-Net over F16 — Constructive and digital
Digital (15, 21, 21846)-net over F16, using
- net defined by OOA [i] based on linear OOA(1621, 21846, F16, 6, 6) (dual of [(21846, 6), 131055, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(1621, 65538, F16, 6) (dual of [65538, 65517, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(1621, 65540, F16, 6) (dual of [65540, 65519, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(1621, 65536, F16, 6) (dual of [65536, 65515, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1617, 65536, F16, 5) (dual of [65536, 65519, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(1621, 65540, F16, 6) (dual of [65540, 65519, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(1621, 65538, F16, 6) (dual of [65538, 65517, 7]-code), using
(15, 21, 65540)-Net over F16 — Digital
Digital (15, 21, 65540)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1621, 65540, F16, 6) (dual of [65540, 65519, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(1621, 65536, F16, 6) (dual of [65536, 65515, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1617, 65536, F16, 5) (dual of [65536, 65519, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
(15, 21, large)-Net in Base 16 — Upper bound on s
There is no (15, 21, large)-net in base 16, because
- 4 times m-reduction [i] would yield (15, 17, large)-net in base 16, but